id	sid	tid	token	lemma	pos
cana-5209	1	1	communications	communication	NOUN
cana-5209	1	2	on	on	ADP
cana-5209	1	3	applied	apply	VERB
cana-5209	1	4	nonlinear	nonlinear	ADJ
cana-5209	1	5	analysis	analysis	NOUN
cana-5209	1	6	issn	issn	NOUN
cana-5209	1	7	:	:	PUNCT
cana-5209	1	8	1074	1074	NUM
cana-5209	1	9	-	-	PUNCT
cana-5209	1	10	133x	133x	NUM
cana-5209	1	11	vol	vol	VERB
cana-5209	1	12	32	32	NUM
cana-5209	1	13	no	no	NOUN
cana-5209	1	14	.	.	PUNCT
cana-5209	2	1	10s	10	NOUN
cana-5209	2	2	(	(	PUNCT
cana-5209	2	3	2025	2025	NUM
cana-5209	2	4	)	)	PUNCT
cana-5209	2	5	optimality	optimality	NOUN
cana-5209	2	6	and	and	CCONJ
cana-5209	2	7	duality	duality	NOUN
cana-5209	2	8	for	for	ADP
cana-5209	2	9	multiobjective	multiobjective	ADJ
cana-5209	2	10	semiinfinite	semiinfinite	ADJ
cana-5209	2	11	programming	programming	NOUN
cana-5209	2	12	problems	problem	NOUN
cana-5209	2	13	involving	involve	VERB
cana-5209	2	14	generalized	generalized	ADJ
cana-5209	2	15	(	(	PUNCT
cana-5209	2	16	c	c	X
cana-5209	2	17	,	,	PUNCT
cana-5209	2	18	α	α	PROPN
cana-5209	2	19	,	,	PUNCT
cana-5209	2	20	η	η	PROPN
cana-5209	2	21	,	,	PUNCT
cana-5209	2	22	ρ	ρ	PROPN
cana-5209	2	23	,	,	PUNCT
cana-5209	2	24	d)-invexity	d)-invexity	NOUN
cana-5209	2	25	pooja	pooja	PROPN
cana-5209	2	26	gupta	gupta	PROPN
cana-5209	2	27	department	department	PROPN
cana-5209	2	28	of	of	ADP
cana-5209	2	29	basic	basic	ADJ
cana-5209	2	30	science	science	NOUN
cana-5209	2	31	&	&	CCONJ
cana-5209	2	32	humanities	humanities	PROPN
cana-5209	2	33	maharana	maharana	PROPN
cana-5209	2	34	pratap	pratap	PROPN
cana-5209	2	35	group	group	PROPN
cana-5209	2	36	of	of	ADP
cana-5209	2	37	institution	institution	NOUN
cana-5209	2	38	,	,	PUNCT
cana-5209	2	39	kanpur	kanpur	PROPN
cana-5209	2	40	,	,	PUNCT
cana-5209	2	41	india	india	PROPN
cana-5209	2	42	email	email	NOUN
cana-5209	2	43	:	:	PUNCT
cana-5209	2	44	pujjubhumath@gmail.com	pujjubhumath@gmail.com	X
cana-5209	2	45	alka	alka	PROPN
cana-5209	2	46	katiyar	katiyar	PROPN
cana-5209	2	47	department	department	PROPN
cana-5209	2	48	of	of	ADP
cana-5209	2	49	mathematics	mathematic	NOUN
cana-5209	2	50	,	,	PUNCT
cana-5209	2	51	school	school	NOUN
cana-5209	2	52	of	of	ADP
cana-5209	2	53	basic	basic	ADJ
cana-5209	2	54	sciences	sciences	PROPN
cana-5209	2	55	csjm	csjm	VERB
cana-5209	2	56	university	university	PROPN
cana-5209	2	57	,	,	PUNCT
cana-5209	2	58	kanpur	kanpur	PROPN
cana-5209	2	59	,	,	PUNCT
cana-5209	2	60	india	india	PROPN
cana-5209	2	61	email:alkakatiyar0092@gmail.com	email:alkakatiyar0092@gmail.com	PROPN
cana-5209	2	62	sandeep	sandeep	PROPN
cana-5209	2	63	kumar	kumar	PROPN
cana-5209	2	64	porwal∗	porwal∗	PROPN
cana-5209	2	65	department	department	PROPN
cana-5209	2	66	of	of	ADP
cana-5209	2	67	mathematics	mathematic	NOUN
cana-5209	2	68	,	,	PUNCT
cana-5209	2	69	school	school	NOUN
cana-5209	2	70	of	of	ADP
cana-5209	2	71	basic	basic	ADJ
cana-5209	2	72	sciences	sciences	PROPN
cana-5209	2	73	csjm	csjm	VERB
cana-5209	2	74	university	university	PROPN
cana-5209	2	75	,	,	PUNCT
cana-5209	2	76	kanpur	kanpur	PROPN
cana-5209	2	77	,	,	PUNCT
cana-5209	2	78	india	india	PROPN
cana-5209	2	79	email:skpmathsdstcims@gmail.com	email:skpmathsdstcims@gmail.com	PROPN
cana-5209	2	80	∗	∗	VERB
cana-5209	2	81	corresponding	correspond	VERB
cana-5209	2	82	author	author	NOUN
cana-5209	2	83	(	(	PUNCT
cana-5209	2	84	article	article	NOUN
cana-5209	2	85	history	history	NOUN
cana-5209	2	86	:	:	PUNCT
cana-5209	2	87	received	receive	VERB
cana-5209	2	88	:	:	PUNCT
cana-5209	2	89	12	12	NUM
cana-5209	2	90	-	-	SYM
cana-5209	2	91	01	01	NUM
cana-5209	2	92	-	-	PUNCT
cana-5209	2	93	2025	2025	NUM
cana-5209	2	94	;	;	PUNCT
cana-5209	2	95	revised	revise	VERB
cana-5209	2	96	:	:	PUNCT
cana-5209	2	97	25	25	NUM
cana-5209	2	98	-	-	PUNCT
cana-5209	2	99	02	02	NUM
cana-5209	2	100	-	-	PUNCT
cana-5209	2	101	2025	2025	NUM
cana-5209	2	102	;	;	PUNCT
cana-5209	2	103	accepted	accept	VERB
cana-5209	2	104	:	:	PUNCT
cana-5209	2	105	05	05	NUM
cana-5209	2	106	-	-	PUNCT
cana-5209	2	107	03	03	NUM
cana-5209	2	108	-	-	PUNCT
cana-5209	2	109	2025	2025	NUM
cana-5209	2	110	)	)	PUNCT
cana-5209	3	1	abstract	abstract	ADV
cana-5209	3	2	in	in	ADP
cana-5209	3	3	this	this	DET
cana-5209	3	4	paper	paper	NOUN
cana-5209	3	5	,	,	PUNCT
cana-5209	3	6	we	we	PRON
cana-5209	3	7	formulate	formulate	VERB
cana-5209	3	8	generalized	generalized	ADJ
cana-5209	3	9	(	(	PUNCT
cana-5209	3	10	c	c	X
cana-5209	3	11	,	,	PUNCT
cana-5209	3	12	α	α	PROPN
cana-5209	3	13	,	,	PUNCT
cana-5209	3	14	η	η	PROPN
cana-5209	3	15	,	,	PUNCT
cana-5209	3	16	ρ	ρ	PROPN
cana-5209	3	17	,	,	PUNCT
cana-5209	3	18	d)-invexity	d)-invexity	NOUN
cana-5209	3	19	and	and	CCONJ
cana-5209	3	20	based	base	VERB
cana-5209	3	21	on	on	ADP
cana-5209	3	22	these	these	DET
cana-5209	3	23	definitions	definition	NOUN
cana-5209	3	24	,	,	PUNCT
cana-5209	3	25	we	we	PRON
cana-5209	3	26	derive	derive	VERB
cana-5209	3	27	several	several	ADJ
cana-5209	3	28	sufficient	sufficient	ADJ
cana-5209	3	29	conditions	condition	NOUN
cana-5209	3	30	for	for	ADP
cana-5209	3	31	optimality	optimality	NOUN
cana-5209	3	32	in	in	ADP
cana-5209	3	33	multiobjective	multiobjective	ADJ
cana-5209	3	34	semi	semi	ADJ
cana-5209	3	35	-	-	ADJ
cana-5209	3	36	infinite	infinite	ADJ
cana-5209	3	37	programming	programming	NOUN
cana-5209	3	38	problems	problem	NOUN
cana-5209	3	39	.	.	PUNCT
cana-5209	4	1	further	far	ADV
cana-5209	4	2	,	,	PUNCT
cana-5209	4	3	under	under	ADP
cana-5209	4	4	the	the	DET
cana-5209	4	5	assumptions	assumption	NOUN
cana-5209	4	6	of	of	ADP
cana-5209	4	7	dual	dual	ADJ
cana-5209	4	8	model	model	NOUN
cana-5209	4	9	we	we	PRON
cana-5209	4	10	solve	solve	VERB
cana-5209	4	11	corresponding	correspond	VERB
cana-5209	4	12	weak	weak	ADJ
cana-5209	4	13	,	,	PUNCT
cana-5209	4	14	strong	strong	ADJ
cana-5209	4	15	and	and	CCONJ
cana-5209	4	16	strict	strict	ADJ
cana-5209	4	17	converse	converse	NOUN
cana-5209	4	18	duality	duality	NOUN
cana-5209	4	19	theorems	theorem	VERB
cana-5209	4	20	for	for	ADP
cana-5209	4	21	these	these	DET
cana-5209	4	22	multiobjective	multiobjective	ADJ
cana-5209	4	23	semiinfinite	semiinfinite	ADJ
cana-5209	4	24	programming	programming	NOUN
cana-5209	4	25	problem	problem	NOUN
cana-5209	4	26	.	.	PUNCT
cana-5209	5	1	2020	2020	NUM
cana-5209	5	2	mathematics	mathematic	NOUN
cana-5209	5	3	subject	subject	ADJ
cana-5209	5	4	classification	classification	NOUN
cana-5209	5	5	:	:	PUNCT
cana-5209	5	6	90c25	90c25	NUM
cana-5209	5	7	,	,	PUNCT
cana-5209	5	8	90c29	90c29	NUM
cana-5209	5	9	,	,	PUNCT
cana-5209	5	10	90c30	90c30	NUM
cana-5209	5	11	,	,	PUNCT
cana-5209	5	12	90c34	90c34	NUM
cana-5209	5	13	,	,	PUNCT
cana-5209	5	14	90c46	90c46	NUM
cana-5209	5	15	keywords	keyword	NOUN
cana-5209	5	16	:	:	PUNCT
cana-5209	5	17	invex	invex	PROPN
cana-5209	5	18	set	set	PROPN
cana-5209	5	19	,	,	PUNCT
cana-5209	5	20	invex	invex	PROPN
cana-5209	5	21	function	function	NOUN
cana-5209	5	22	,	,	PUNCT
cana-5209	5	23	multiobjective	multiobjective	ADJ
cana-5209	5	24	programming	programming	NOUN
cana-5209	5	25	,	,	PUNCT
cana-5209	5	26	semiinfinite	semiinfinite	NOUN
cana-5209	5	27	programming	programming	NOUN
cana-5209	5	28	,	,	PUNCT
cana-5209	5	29	feasible	feasible	ADJ
cana-5209	5	30	solution	solution	NOUN
cana-5209	5	31	,	,	PUNCT
cana-5209	5	32	efficient	efficient	ADJ
cana-5209	5	33	solution	solution	NOUN
cana-5209	5	34	.	.	PUNCT
cana-5209	6	1	1	1	NUM
cana-5209	6	2	introduction	introduction	NOUN
cana-5209	6	3	the	the	DET
cana-5209	6	4	theory	theory	NOUN
cana-5209	6	5	surrounding	surround	VERB
cana-5209	6	6	semi	semi	ADJ
cana-5209	6	7	-	-	ADJ
cana-5209	6	8	infinite	infinite	ADJ
cana-5209	6	9	programming	programming	NOUN
cana-5209	6	10	involves	involve	VERB
cana-5209	6	11	minimizing	minimize	VERB
cana-5209	6	12	a	a	DET
cana-5209	6	13	function	function	NOUN
cana-5209	6	14	with	with	ADP
cana-5209	6	15	a	a	DET
cana-5209	6	16	finite	finite	ADJ
cana-5209	6	17	number	number	NOUN
cana-5209	6	18	of	of	ADP
cana-5209	6	19	variables	variable	NOUN
cana-5209	6	20	while	while	SCONJ
cana-5209	6	21	adhering	adhere	VERB
cana-5209	6	22	to	to	ADP
cana-5209	6	23	an	an	DET
cana-5209	6	24	arbitrary	arbitrary	ADJ
cana-5209	6	25	number	number	NOUN
cana-5209	6	26	of	of	ADP
cana-5209	6	27	inequalities	inequality	NOUN
cana-5209	6	28	.	.	PUNCT
cana-5209	7	1	when	when	SCONJ
cana-5209	7	2	there	there	PRON
cana-5209	7	3	are	be	VERB
cana-5209	7	4	multiple	multiple	ADJ
cana-5209	7	5	objective	objective	ADJ
cana-5209	7	6	functions	function	NOUN
cana-5209	7	7	,	,	PUNCT
cana-5209	7	8	this	this	PRON
cana-5209	7	9	is	be	AUX
cana-5209	7	10	referred	refer	VERB
cana-5209	7	11	to	to	ADP
cana-5209	7	12	as	as	ADP
cana-5209	7	13	a	a	DET
cana-5209	7	14	multiobjective	multiobjective	ADJ
cana-5209	7	15	semi	semi	ADJ
cana-5209	7	16	-	-	ADJ
cana-5209	7	17	infinite	infinite	ADJ
cana-5209	7	18	programming	programming	NOUN
cana-5209	7	19	problem	problem	NOUN
cana-5209	7	20	.	.	PUNCT
cana-5209	8	1	shapiro	shapiro	PROPN
cana-5209	9	1	[	[	X
cana-5209	9	2	7	7	NUM
cana-5209	9	3	]	]	PUNCT
cana-5209	9	4	provided	provide	VERB
cana-5209	9	5	an	an	DET
cana-5209	9	6	overview	overview	NOUN
cana-5209	9	7	of	of	ADP
cana-5209	9	8	the	the	DET
cana-5209	9	9	foundational	foundational	ADJ
cana-5209	9	10	theory	theory	NOUN
cana-5209	9	11	of	of	ADP
cana-5209	9	12	semi	semi	ADJ
cana-5209	9	13	-	-	ADJ
cana-5209	9	14	infinite	infinite	ADJ
cana-5209	9	15	programming	programming	NOUN
cana-5209	9	16	,	,	PUNCT
cana-5209	9	17	exploring	explore	VERB
cana-5209	9	18	various	various	ADJ
cana-5209	9	19	methods	method	NOUN
cana-5209	9	20	for	for	ADP
cana-5209	9	21	establishing	establish	VERB
cana-5209	9	22	duality	duality	NOUN
cana-5209	9	23	,	,	PUNCT
cana-5209	9	24	discretization	discretization	NOUN
cana-5209	9	25	,	,	PUNCT
cana-5209	9	26	and	and	CCONJ
cana-5209	9	27	both	both	DET
cana-5209	9	28	first	first	ADJ
cana-5209	9	29	and	and	CCONJ
cana-5209	9	30	second	second	ADJ
cana-5209	9	31	-	-	PUNCT
cana-5209	9	32	order	order	NOUN
cana-5209	9	33	optimality	optimality	NOUN
cana-5209	9	34	conditions	condition	NOUN
cana-5209	9	35	.	.	PUNCT
cana-5209	10	1	for	for	ADP
cana-5209	10	2	further	further	ADJ
cana-5209	10	3	information	information	NOUN
cana-5209	10	4	and	and	CCONJ
cana-5209	10	5	applications	application	NOUN
cana-5209	10	6	related	relate	VERB
cana-5209	10	7	to	to	ADP
cana-5209	10	8	semi	semi	ADJ
cana-5209	10	9	-	-	ADJ
cana-5209	10	10	infinite	infinite	ADJ
cana-5209	10	11	programming	programming	NOUN
cana-5209	10	12	,	,	PUNCT
cana-5209	10	13	please	please	INTJ
cana-5209	10	14	consult	consult	VERB
cana-5209	10	15	the	the	DET
cana-5209	10	16	cited	cite	VERB
cana-5209	10	17	references	reference	NOUN
cana-5209	10	18	.	.	PUNCT
cana-5209	11	1	[	[	X
cana-5209	11	2	2	2	NUM
cana-5209	11	3	,	,	PUNCT
cana-5209	11	4	3	3	NUM
cana-5209	11	5	,	,	PUNCT
cana-5209	11	6	4	4	NUM
cana-5209	11	7	,	,	PUNCT
cana-5209	11	8	5	5	NUM
cana-5209	11	9	,	,	PUNCT
cana-5209	11	10	6	6	NUM
cana-5209	11	11	,	,	PUNCT
cana-5209	11	12	8	8	NUM
cana-5209	11	13	,	,	PUNCT
cana-5209	11	14	11	11	NUM
cana-5209	11	15	,	,	PUNCT
cana-5209	11	16	12	12	NUM
cana-5209	11	17	,	,	PUNCT
cana-5209	11	18	33	33	NUM
cana-5209	11	19	,	,	PUNCT
cana-5209	11	20	34	34	NUM
cana-5209	11	21	]	]	PUNCT
cana-5209	11	22	.	.	PUNCT
cana-5209	12	1	in	in	ADP
cana-5209	12	2	[	[	X
cana-5209	12	3	28	28	NUM
cana-5209	12	4	]	]	PUNCT
cana-5209	12	5	,	,	PUNCT
cana-5209	12	6	preda	preda	PROPN
cana-5209	12	7	introduced	introduce	VERB
cana-5209	12	8	the	the	DET
cana-5209	12	9	idea	idea	NOUN
cana-5209	12	10	of	of	ADP
cana-5209	12	11	(	(	PUNCT
cana-5209	12	12	f	f	X
cana-5209	12	13	,	,	PUNCT
cana-5209	12	14	ρ)-convexity	ρ)-convexity	NOUN
cana-5209	12	15	,	,	PUNCT
cana-5209	12	16	expanding	expand	VERB
cana-5209	12	17	upon	upon	SCONJ
cana-5209	12	18	the	the	DET
cana-5209	12	19	concepts	concept	NOUN
cana-5209	12	20	of	of	ADP
cana-5209	12	21	f	f	PROPN
cana-5209	12	22	-convexity	-convexity	PROPN
cana-5209	13	1	[	[	X
cana-5209	13	2	18	18	NUM
cana-5209	13	3	]	]	PUNCT
cana-5209	13	4	and	and	CCONJ
cana-5209	13	5	ρ	ρ	PROPN
cana-5209	13	6	-	-	NOUN
cana-5209	13	7	convexity	convexity	NOUN
cana-5209	14	1	[	[	X
cana-5209	14	2	16	16	NUM
cana-5209	14	3	]	]	PUNCT
cana-5209	14	4	,	,	PUNCT
cana-5209	14	5	and	and	CCONJ
cana-5209	14	6	derived	derive	VERB
cana-5209	14	7	several	several	ADJ
cana-5209	14	8	duality	duality	NOUN
cana-5209	14	9	results	result	NOUN
cana-5209	14	10	.	.	PUNCT
cana-5209	15	1	liang	liang	PROPN
cana-5209	15	2	et	et	PROPN
cana-5209	15	3	al	al	PROPN
cana-5209	15	4	.	.	PUNCT
cana-5209	16	1	in	in	ADP
cana-5209	16	2	[	[	X
cana-5209	16	3	31	31	NUM
cana-5209	16	4	]	]	PUNCT
cana-5209	16	5	later	later	ADV
cana-5209	16	6	presented	present	VERB
cana-5209	16	7	(	(	PUNCT
cana-5209	16	8	f	f	X
cana-5209	16	9	,	,	PUNCT
cana-5209	16	10	α	α	PROPN
cana-5209	16	11	,	,	PUNCT
cana-5209	16	12	ρ	ρ	PROPN
cana-5209	16	13	,	,	PUNCT
cana-5209	16	14	d)-convexity	d)-convexity	NOUN
cana-5209	16	15	to	to	PART
cana-5209	16	16	address	address	VERB
cana-5209	16	17	nonlinear	nonlinear	ADJ
cana-5209	16	18	fractional	fractional	ADJ
cana-5209	16	19	programming	programming	NOUN
cana-5209	16	20	problems	problem	NOUN
cana-5209	16	21	,	,	PUNCT
cana-5209	16	22	which	which	PRON
cana-5209	16	23	encompasses	encompass	VERB
cana-5209	16	24	the	the	DET
cana-5209	16	25	(	(	PUNCT
cana-5209	16	26	f	f	NUM
cana-5209	16	27	,	,	PUNCT
cana-5209	16	28	ρ)-convex	ρ)-convex	NOUN
cana-5209	16	29	functions	function	NOUN
cana-5209	16	30	.	.	PUNCT
cana-5209	17	1	subsequently	subsequently	ADV
cana-5209	17	2	,	,	PUNCT
cana-5209	17	3	liang	liang	PROPN
cana-5209	17	4	et	et	PROPN
cana-5209	17	5	al	al	PROPN
cana-5209	17	6	.	.	PUNCT
cana-5209	18	1	in	in	ADP
cana-5209	18	2	[	[	X
cana-5209	18	3	30	30	NUM
cana-5209	18	4	]	]	PUNCT
cana-5209	18	5	broadened	broaden	VERB
cana-5209	18	6	the	the	DET
cana-5209	18	7	findings	finding	NOUN
cana-5209	18	8	from	from	ADP
cana-5209	18	9	[	[	X
cana-5209	18	10	31	31	NUM
cana-5209	18	11	]	]	PUNCT
cana-5209	18	12	to	to	PART
cana-5209	18	13	include	include	VERB
cana-5209	18	14	a	a	DET
cana-5209	18	15	specific	specific	ADJ
cana-5209	18	16	category	category	NOUN
cana-5209	18	17	of	of	ADP
cana-5209	18	18	multiobjective	multiobjective	ADJ
cana-5209	18	19	fractional	fractional	ADJ
cana-5209	18	20	programming	programming	NOUN
cana-5209	18	21	problems	problem	NOUN
cana-5209	18	22	.	.	PUNCT
cana-5209	19	1	yuan	yuan	NOUN
cana-5209	19	2	et	et	PROPN
cana-5209	19	3	al	al	PROPN
cana-5209	19	4	.	.	PROPN
cana-5209	19	5	defined	define	VERB
cana-5209	19	6	(	(	PUNCT
cana-5209	19	7	c	c	X
cana-5209	19	8	,	,	PUNCT
cana-5209	19	9	α	α	PROPN
cana-5209	19	10	,	,	PUNCT
cana-5209	19	11	ρ	ρ	NOUN
cana-5209	19	12	,	,	PUNCT
cana-5209	19	13	d)-convexity	d)-convexity	NOUN
cana-5209	19	14	in	in	ADP
cana-5209	19	15	[	[	X
cana-5209	19	16	13	13	NUM
cana-5209	19	17	]	]	PUNCT
cana-5209	19	18	,	,	PUNCT
cana-5209	19	19	as	as	ADP
cana-5209	19	20	a	a	DET
cana-5209	19	21	generalization	generalization	NOUN
cana-5209	19	22	of	of	ADP
cana-5209	19	23	(	(	PUNCT
cana-5209	19	24	f	f	PROPN
cana-5209	19	25	,	,	PUNCT
cana-5209	19	26	α	α	PROPN
cana-5209	19	27	,	,	PUNCT
cana-5209	19	28	ρ	ρ	NOUN
cana-5209	19	29	,	,	PUNCT
cana-5209	19	30	d)-convexity	d)-convexity	NOUN
cana-5209	19	31	and	and	CCONJ
cana-5209	19	32	established	establish	VERB
cana-5209	19	33	optimality	optimality	NOUN
cana-5209	19	34	conditions	condition	NOUN
cana-5209	19	35	along	along	ADP
cana-5209	19	36	with	with	ADP
cana-5209	19	37	duality	duality	NOUN
cana-5209	19	38	results	result	NOUN
cana-5209	19	39	for	for	ADP
cana-5209	19	40	nondifferentiable	nondifferentiable	ADJ
cana-5209	19	41	minimax	minimax	NOUN
cana-5209	19	42	fractional	fractional	ADJ
cana-5209	19	43	programming	programming	NOUN
cana-5209	19	44	problems	problem	NOUN
cana-5209	19	45	that	that	PRON
cana-5209	19	46	utilize	utilize	VERB
cana-5209	19	47	generalized	generalized	ADJ
cana-5209	19	48	convex	convex	NOUN
cana-5209	19	49	functions	function	NOUN
cana-5209	19	50	.	.	PUNCT
cana-5209	20	1	additionally	additionally	ADV
cana-5209	20	2	,	,	PUNCT
cana-5209	20	3	long	long	ADV
cana-5209	20	4	in	in	ADP
cana-5209	20	5	[	[	X
cana-5209	20	6	29	29	NUM
cana-5209	20	7	]	]	PUNCT
cana-5209	20	8	and	and	CCONJ
cana-5209	20	9	mishra	mishra	PROPN
cana-5209	20	10	et	et	PROPN
cana-5209	20	11	al	al	PROPN
cana-5209	20	12	.	.	PUNCT
cana-5209	21	1	in	in	ADP
cana-5209	21	2	[	[	X
cana-5209	21	3	25	25	NUM
cana-5209	21	4	]	]	PUNCT
cana-5209	21	5	derived	derive	VERB
cana-5209	21	6	sufficient	sufficient	ADJ
cana-5209	21	7	optimality	optimality	NOUN
cana-5209	21	8	conditions	condition	NOUN
cana-5209	21	9	and	and	CCONJ
cana-5209	21	10	duality	duality	NOUN
cana-5209	21	11	theorems	theorem	NOUN
cana-5209	21	12	employing	employ	VERB
cana-5209	21	13	(	(	PUNCT
cana-5209	21	14	c	c	NOUN
cana-5209	21	15	,	,	PUNCT
cana-5209	21	16	α	α	PROPN
cana-5209	21	17	,	,	PUNCT
cana-5209	21	18	ρ	ρ	NOUN
cana-5209	21	19	,	,	PUNCT
cana-5209	21	20	d)-convexity	d)-convexity	NOUN
cana-5209	21	21	for	for	ADP
cana-5209	21	22	nondifferentiable	nondifferentiable	ADJ
cana-5209	21	23	multiobjective	multiobjective	ADJ
cana-5209	21	24	fractional	fractional	ADJ
cana-5209	21	25	programming	programming	NOUN
cana-5209	21	26	and	and	CCONJ
cana-5209	21	27	nondifferentiable	nondifferentiable	ADJ
cana-5209	21	28	multiobjective	multiobjective	ADJ
cana-5209	21	29	semi	semi	ADJ
cana-5209	21	30	-	-	ADJ
cana-5209	21	31	infinite	infinite	ADJ
cana-5209	21	32	https://internationalpubls.com	https://internationalpubls.com	X
cana-5209	21	33	1140	1140	NUM
cana-5209	21	34	communications	communication	NOUN
cana-5209	21	35	on	on	ADP
cana-5209	21	36	applied	apply	VERB
cana-5209	21	37	nonlinear	nonlinear	ADJ
cana-5209	21	38	analysis	analysis	NOUN
cana-5209	21	39	issn	issn	NOUN
cana-5209	21	40	:	:	PUNCT
cana-5209	21	41	1074	1074	NUM
cana-5209	21	42	-	-	PUNCT
cana-5209	21	43	133x	133x	NUM
cana-5209	21	44	vol	vol	VERB
cana-5209	21	45	32	32	NUM
cana-5209	21	46	no	no	NOUN
cana-5209	21	47	.	.	PUNCT
cana-5209	22	1	10s	10	NOUN
cana-5209	22	2	(	(	PUNCT
cana-5209	22	3	2025	2025	NUM
cana-5209	22	4	)	)	PUNCT
cana-5209	22	5	programming	programming	NOUN
cana-5209	22	6	problems	problem	NOUN
cana-5209	22	7	,	,	PUNCT
cana-5209	22	8	respectively	respectively	ADV
cana-5209	22	9	.	.	PUNCT
cana-5209	23	1	mishra	mishra	PROPN
cana-5209	23	2	et	et	PROPN
cana-5209	23	3	al	al	PROPN
cana-5209	23	4	.	.	PUNCT
cana-5209	24	1	[	[	X
cana-5209	24	2	21	21	NUM
cana-5209	24	3	]	]	PUNCT
cana-5209	24	4	also	also	ADV
cana-5209	24	5	achieved	achieve	VERB
cana-5209	24	6	optimality	optimality	NOUN
cana-5209	24	7	and	and	CCONJ
cana-5209	24	8	duality	duality	NOUN
cana-5209	24	9	results	result	NOUN
cana-5209	24	10	for	for	ADP
cana-5209	24	11	minimax	minimax	NOUN
cana-5209	24	12	fractional	fractional	ADJ
cana-5209	24	13	programming	programming	NOUN
cana-5209	24	14	involving	involve	VERB
cana-5209	24	15	support	support	NOUN
cana-5209	24	16	functions	function	NOUN
cana-5209	24	17	within	within	ADP
cana-5209	24	18	the	the	DET
cana-5209	24	19	framework	framework	NOUN
cana-5209	24	20	of	of	ADP
cana-5209	24	21	(	(	PUNCT
cana-5209	24	22	c	c	PROPN
cana-5209	24	23	,	,	PUNCT
cana-5209	24	24	α	α	PROPN
cana-5209	24	25	,	,	PUNCT
cana-5209	24	26	ρ	ρ	NOUN
cana-5209	24	27	,	,	PUNCT
cana-5209	24	28	d)-convexity	d)-convexity	NOUN
cana-5209	24	29	.	.	PUNCT
cana-5209	25	1	invexity	invexity	NOUN
cana-5209	25	2	is	be	AUX
cana-5209	25	3	crucial	crucial	ADJ
cana-5209	25	4	for	for	ADP
cana-5209	25	5	establishing	establish	VERB
cana-5209	25	6	optimality	optimality	NOUN
cana-5209	25	7	conditions	condition	NOUN
cana-5209	25	8	and	and	CCONJ
cana-5209	25	9	duality	duality	NOUN
cana-5209	25	10	results	result	VERB
cana-5209	25	11	across	across	ADP
cana-5209	25	12	a	a	DET
cana-5209	25	13	range	range	NOUN
cana-5209	25	14	of	of	ADP
cana-5209	25	15	optimization	optimization	NOUN
cana-5209	25	16	problems	problem	NOUN
cana-5209	25	17	.	.	PUNCT
cana-5209	26	1	invex	invex	PROPN
cana-5209	26	2	functions	function	NOUN
cana-5209	26	3	represent	represent	VERB
cana-5209	26	4	a	a	DET
cana-5209	26	5	broader	broad	ADJ
cana-5209	26	6	category	category	NOUN
cana-5209	26	7	than	than	ADP
cana-5209	26	8	convex	convex	NOUN
cana-5209	26	9	functions	function	NOUN
cana-5209	26	10	,	,	PUNCT
cana-5209	26	11	retaining	retain	VERB
cana-5209	26	12	many	many	ADJ
cana-5209	26	13	of	of	ADP
cana-5209	26	14	their	their	PRON
cana-5209	26	15	key	key	ADJ
cana-5209	26	16	properties	property	NOUN
cana-5209	26	17	.	.	PUNCT
cana-5209	27	1	additionally	additionally	ADV
cana-5209	27	2	,	,	PUNCT
cana-5209	27	3	invexity	invexity	NOUN
cana-5209	27	4	and	and	CCONJ
cana-5209	27	5	its	its	PRON
cana-5209	27	6	generalizations	generalization	NOUN
cana-5209	27	7	can	can	AUX
cana-5209	27	8	be	be	AUX
cana-5209	27	9	viewed	view	VERB
cana-5209	27	10	as	as	ADP
cana-5209	27	11	alternatives	alternative	NOUN
cana-5209	27	12	to	to	ADP
cana-5209	27	13	convexity	convexity	NOUN
cana-5209	27	14	.	.	PUNCT
cana-5209	28	1	for	for	ADP
cana-5209	28	2	further	further	ADJ
cana-5209	28	3	information	information	NOUN
cana-5209	28	4	on	on	ADP
cana-5209	28	5	invex	invex	NOUN
cana-5209	28	6	functions	function	NOUN
cana-5209	28	7	,	,	PUNCT
cana-5209	28	8	please	please	INTJ
cana-5209	28	9	refer	refer	VERB
cana-5209	28	10	to	to	ADP
cana-5209	28	11	the	the	DET
cana-5209	28	12	relevant	relevant	ADJ
cana-5209	28	13	literature	literature	NOUN
cana-5209	28	14	[	[	X
cana-5209	28	15	1	1	NUM
cana-5209	28	16	,	,	PUNCT
cana-5209	28	17	9	9	NUM
cana-5209	28	18	,	,	PUNCT
cana-5209	28	19	10	10	NUM
cana-5209	28	20	,	,	PUNCT
cana-5209	28	21	17	17	NUM
cana-5209	28	22	,	,	PUNCT
cana-5209	28	23	22	22	NUM
cana-5209	28	24	,	,	PUNCT
cana-5209	28	25	23	23	NUM
cana-5209	28	26	,	,	PUNCT
cana-5209	28	27	32	32	NUM
cana-5209	28	28	]	]	PUNCT
cana-5209	28	29	.	.	PUNCT
cana-5209	29	1	weir	weir	PROPN
cana-5209	30	1	[	[	X
cana-5209	30	2	27	27	NUM
cana-5209	30	3	]	]	PUNCT
cana-5209	30	4	examined	examine	VERB
cana-5209	30	5	a	a	DET
cana-5209	30	6	multiobjective	multiobjective	ADJ
cana-5209	30	7	programming	programming	NOUN
cana-5209	30	8	problem	problem	NOUN
cana-5209	30	9	that	that	PRON
cana-5209	30	10	incorporates	incorporate	VERB
cana-5209	30	11	invex	invex	NOUN
cana-5209	30	12	functions	function	NOUN
cana-5209	30	13	and	and	CCONJ
cana-5209	30	14	derived	derive	VERB
cana-5209	30	15	duality	duality	NOUN
cana-5209	30	16	results	result	NOUN
cana-5209	30	17	under	under	ADP
cana-5209	30	18	the	the	DET
cana-5209	30	19	condition	condition	NOUN
cana-5209	30	20	that	that	SCONJ
cana-5209	30	21	the	the	DET
cana-5209	30	22	multipliers	multiplier	NOUN
cana-5209	30	23	for	for	ADP
cana-5209	30	24	all	all	DET
cana-5209	30	25	objective	objective	ADJ
cana-5209	30	26	functions	function	NOUN
cana-5209	30	27	are	be	AUX
cana-5209	30	28	strictly	strictly	ADV
cana-5209	30	29	positive	positive	ADJ
cana-5209	30	30	.	.	PUNCT
cana-5209	31	1	kuk	kuk	PROPN
cana-5209	31	2	et	et	PROPN
cana-5209	31	3	al	al	PROPN
cana-5209	31	4	.	.	PUNCT
cana-5209	32	1	in	in	ADP
cana-5209	32	2	[	[	X
cana-5209	32	3	15	15	NUM
cana-5209	32	4	]	]	PUNCT
cana-5209	32	5	formulated	formulate	VERB
cana-5209	32	6	generalized	generalized	ADJ
cana-5209	32	7	k	k	PROPN
cana-5209	32	8	-	-	PUNCT
cana-5209	32	9	k	k	NOUN
cana-5209	32	10	-	-	PUNCT
cana-5209	32	11	t	t	NOUN
cana-5209	32	12	necessary	necessary	ADJ
cana-5209	32	13	and	and	CCONJ
cana-5209	32	14	sufficient	sufficient	ADJ
cana-5209	32	15	optimality	optimality	NOUN
cana-5209	32	16	conditions	condition	NOUN
cana-5209	32	17	,	,	PUNCT
cana-5209	32	18	along	along	ADP
cana-5209	32	19	with	with	ADP
cana-5209	32	20	duality	duality	NOUN
cana-5209	32	21	theorems	theorem	NOUN
cana-5209	32	22	,	,	PUNCT
cana-5209	32	23	for	for	ADP
cana-5209	32	24	nonsmooth	nonsmooth	ADJ
cana-5209	32	25	multiobjective	multiobjective	ADJ
cana-5209	32	26	fractional	fractional	ADJ
cana-5209	32	27	programming	programming	NOUN
cana-5209	32	28	problems	problem	NOUN
cana-5209	32	29	that	that	PRON
cana-5209	32	30	involve	involve	VERB
cana-5209	32	31	v	v	PRON
cana-5209	32	32	−ρ−	−ρ−	NOUN
cana-5209	32	33	invex	invex	NOUN
cana-5209	32	34	functions	function	NOUN
cana-5209	32	35	.	.	PUNCT
cana-5209	33	1	caristi	caristi	PROPN
cana-5209	33	2	et	et	PROPN
cana-5209	33	3	al	al	PROPN
cana-5209	33	4	.	.	PUNCT
cana-5209	34	1	[	[	X
cana-5209	34	2	14	14	NUM
cana-5209	34	3	]	]	PUNCT
cana-5209	34	4	focused	focus	VERB
cana-5209	34	5	on	on	ADP
cana-5209	34	6	multiobjective	multiobjective	ADJ
cana-5209	34	7	programming	programming	NOUN
cana-5209	34	8	with	with	ADP
cana-5209	34	9	a	a	DET
cana-5209	34	10	set	set	NOUN
cana-5209	34	11	of	of	ADP
cana-5209	34	12	constraints	constraint	NOUN
cana-5209	34	13	defined	define	VERB
cana-5209	34	14	within	within	ADP
cana-5209	34	15	a	a	DET
cana-5209	34	16	compact	compact	ADJ
cana-5209	34	17	set	set	NOUN
cana-5209	34	18	,	,	PUNCT
cana-5209	34	19	obtaining	obtain	VERB
cana-5209	34	20	kuhn	kuhn	PROPN
cana-5209	34	21	-	-	PUNCT
cana-5209	34	22	tucker	tucker	PROPN
cana-5209	34	23	type	type	NOUN
cana-5209	34	24	optimality	optimality	NOUN
cana-5209	34	25	criteria	criterion	NOUN
cana-5209	34	26	under	under	ADP
cana-5209	34	27	relaxed	relaxed	ADJ
cana-5209	34	28	invexity	invexity	NOUN
cana-5209	34	29	conditions	condition	NOUN
cana-5209	34	30	.	.	PUNCT
cana-5209	35	1	they	they	PRON
cana-5209	35	2	also	also	ADV
cana-5209	35	3	introduced	introduce	VERB
cana-5209	35	4	dual	dual	ADJ
cana-5209	35	5	problems	problem	NOUN
cana-5209	35	6	where	where	SCONJ
cana-5209	35	7	both	both	CCONJ
cana-5209	35	8	weak	weak	ADJ
cana-5209	35	9	and	and	CCONJ
cana-5209	35	10	strong	strong	ADJ
cana-5209	35	11	duality	duality	NOUN
cana-5209	35	12	properties	property	NOUN
cana-5209	35	13	are	be	AUX
cana-5209	35	14	maintained	maintain	VERB
cana-5209	35	15	in	in	ADP
cana-5209	35	16	the	the	DET
cana-5209	35	17	same	same	ADJ
cana-5209	35	18	framework	framework	NOUN
cana-5209	35	19	.	.	PUNCT
cana-5209	36	1	additionally	additionally	ADV
cana-5209	36	2	,	,	PUNCT
cana-5209	36	3	mishra	mishra	PROPN
cana-5209	36	4	et	et	PROPN
cana-5209	36	5	al	al	PROPN
cana-5209	36	6	.	.	PUNCT
cana-5209	37	1	[	[	X
cana-5209	37	2	24	24	NUM
cana-5209	37	3	]	]	PUNCT
cana-5209	37	4	developed	develop	VERB
cana-5209	37	5	wolfe	wolfe	PROPN
cana-5209	37	6	and	and	CCONJ
cana-5209	37	7	mond	mond	PROPN
cana-5209	37	8	-	-	PUNCT
cana-5209	37	9	weir	weir	NOUN
cana-5209	37	10	-	-	PUNCT
cana-5209	37	11	type	type	NOUN
cana-5209	37	12	dual	dual	ADJ
cana-5209	37	13	models	model	NOUN
cana-5209	37	14	,	,	PUNCT
cana-5209	37	15	establishing	establish	VERB
cana-5209	37	16	duality	duality	NOUN
cana-5209	37	17	theorems	theorem	NOUN
cana-5209	37	18	for	for	ADP
cana-5209	37	19	the	the	DET
cana-5209	37	20	nonsmooth	nonsmooth	NOUN
cana-5209	37	21	semi	semi	ADJ
cana-5209	37	22	-	-	ADJ
cana-5209	37	23	infinite	infinite	ADJ
cana-5209	37	24	programming	programming	NOUN
cana-5209	37	25	problem	problem	NOUN
cana-5209	37	26	discussed	discuss	VERB
cana-5209	37	27	in	in	ADP
cana-5209	37	28	[	[	X
cana-5209	37	29	20	20	NUM
cana-5209	37	30	]	]	PUNCT
cana-5209	37	31	.	.	PUNCT
cana-5209	38	1	motivated	motivate	VERB
cana-5209	38	2	by	by	ADP
cana-5209	38	3	the	the	DET
cana-5209	38	4	research	research	NOUN
cana-5209	38	5	of	of	ADP
cana-5209	38	6	ben	ben	PROPN
cana-5209	38	7	-	-	PROPN
cana-5209	38	8	israel	israel	PROPN
cana-5209	38	9	and	and	CCONJ
cana-5209	38	10	mond	mond	VERB
cana-5209	39	1	[	[	X
cana-5209	39	2	1	1	NUM
cana-5209	39	3	]	]	PUNCT
cana-5209	39	4	,	,	PUNCT
cana-5209	39	5	caristi	caristi	PROPN
cana-5209	39	6	et	et	PROPN
cana-5209	39	7	al	al	PROPN
cana-5209	39	8	.	.	PUNCT
cana-5209	40	1	[	[	X
cana-5209	40	2	14	14	NUM
cana-5209	40	3	]	]	PUNCT
cana-5209	40	4	and	and	CCONJ
cana-5209	40	5	jaiswal	jaiswal	PROPN
cana-5209	40	6	and	and	CCONJ
cana-5209	40	7	mishra	mishra	PROPN
cana-5209	40	8	[	[	X
cana-5209	40	9	19	19	NUM
cana-5209	40	10	]	]	PUNCT
cana-5209	40	11	,	,	PUNCT
cana-5209	40	12	we	we	PRON
cana-5209	40	13	extend	extend	VERB
cana-5209	40	14	the	the	DET
cana-5209	40	15	definition	definition	NOUN
cana-5209	40	16	of	of	ADP
cana-5209	40	17	generalized	generalized	ADJ
cana-5209	40	18	(	(	PUNCT
cana-5209	40	19	c	c	X
cana-5209	40	20	,	,	PUNCT
cana-5209	40	21	α	α	PROPN
cana-5209	40	22	,	,	PUNCT
cana-5209	40	23	ρ	ρ	PROPN
cana-5209	40	24	,	,	PUNCT
cana-5209	40	25	d)-convexity	d)-convexity	NOUN
cana-5209	40	26	to	to	PART
cana-5209	40	27	generalized	generalize	VERB
cana-5209	40	28	(	(	PUNCT
cana-5209	40	29	c	c	X
cana-5209	40	30	,	,	PUNCT
cana-5209	40	31	α	α	PROPN
cana-5209	40	32	,	,	PUNCT
cana-5209	40	33	η	η	PROPN
cana-5209	40	34	,	,	PUNCT
cana-5209	40	35	ρ	ρ	PROPN
cana-5209	40	36	,	,	PUNCT
cana-5209	40	37	d)-invexity	d)-invexity	NOUN
cana-5209	40	38	.	.	PUNCT
cana-5209	41	1	additionally	additionally	ADV
cana-5209	41	2	,	,	PUNCT
cana-5209	41	3	we	we	PRON
cana-5209	41	4	present	present	VERB
cana-5209	41	5	sufficient	sufficient	ADJ
cana-5209	41	6	conditions	condition	NOUN
cana-5209	41	7	for	for	ADP
cana-5209	41	8	optimality	optimality	NOUN
cana-5209	41	9	and	and	CCONJ
cana-5209	41	10	duality	duality	NOUN
cana-5209	41	11	theorems	theorem	NOUN
cana-5209	41	12	.	.	PUNCT
cana-5209	42	1	the	the	DET
cana-5209	42	2	structure	structure	NOUN
cana-5209	42	3	of	of	ADP
cana-5209	42	4	this	this	DET
cana-5209	42	5	paper	paper	NOUN
cana-5209	42	6	is	be	AUX
cana-5209	42	7	organized	organize	VERB
cana-5209	42	8	as	as	SCONJ
cana-5209	42	9	follows	follow	VERB
cana-5209	42	10	:	:	PUNCT
cana-5209	42	11	in	in	ADP
cana-5209	42	12	section	section	NOUN
cana-5209	42	13	2	2	NUM
cana-5209	42	14	,	,	PUNCT
cana-5209	42	15	we	we	PRON
cana-5209	42	16	provide	provide	VERB
cana-5209	42	17	some	some	DET
cana-5209	42	18	preliminaries	preliminary	NOUN
cana-5209	42	19	,	,	PUNCT
cana-5209	42	20	our	our	PRON
cana-5209	42	21	problem	problem	NOUN
cana-5209	42	22	and	and	CCONJ
cana-5209	42	23	some	some	DET
cana-5209	42	24	definitions	definition	NOUN
cana-5209	42	25	.	.	PUNCT
cana-5209	43	1	section	section	NOUN
cana-5209	43	2	3	3	NUM
cana-5209	43	3	is	be	AUX
cana-5209	43	4	dedicated	dedicate	VERB
cana-5209	43	5	to	to	ADP
cana-5209	43	6	establishing	establish	VERB
cana-5209	43	7	definitions	definition	NOUN
cana-5209	43	8	that	that	PRON
cana-5209	43	9	will	will	AUX
cana-5209	43	10	be	be	AUX
cana-5209	43	11	essential	essential	ADJ
cana-5209	43	12	for	for	ADP
cana-5209	43	13	our	our	PRON
cana-5209	43	14	theorems	theorem	NOUN
cana-5209	43	15	.	.	PUNCT
cana-5209	44	1	in	in	ADP
cana-5209	44	2	section	section	NOUN
cana-5209	44	3	4	4	NUM
cana-5209	44	4	,	,	PUNCT
cana-5209	44	5	we	we	PRON
cana-5209	44	6	derive	derive	VERB
cana-5209	44	7	sufficient	sufficient	ADJ
cana-5209	44	8	conditions	condition	NOUN
cana-5209	44	9	for	for	ADP
cana-5209	44	10	optimality	optimality	NOUN
cana-5209	44	11	.	.	PUNCT
cana-5209	45	1	finally	finally	ADV
cana-5209	45	2	,	,	PUNCT
cana-5209	45	3	section	section	NOUN
cana-5209	45	4	5	5	NUM
cana-5209	45	5	presents	present	VERB
cana-5209	45	6	weak	weak	ADJ
cana-5209	45	7	,	,	PUNCT
cana-5209	45	8	strong	strong	ADJ
cana-5209	45	9	,	,	PUNCT
cana-5209	45	10	and	and	CCONJ
cana-5209	45	11	strict	strict	ADJ
cana-5209	45	12	converse	converse	NOUN
cana-5209	45	13	duality	duality	NOUN
cana-5209	45	14	theorems	theorem	NOUN
cana-5209	45	15	that	that	PRON
cana-5209	45	16	connect	connect	VERB
cana-5209	45	17	the	the	DET
cana-5209	45	18	primal	primal	ADJ
cana-5209	45	19	problem	problem	NOUN
cana-5209	45	20	with	with	ADP
cana-5209	45	21	the	the	DET
cana-5209	45	22	mond	mond	PROPN
cana-5209	45	23	-	-	PUNCT
cana-5209	45	24	weir	weir	PROPN
cana-5209	45	25	dual	dual	ADJ
cana-5209	45	26	problem	problem	NOUN
cana-5209	45	27	under	under	ADP
cana-5209	45	28	the	the	DET
cana-5209	45	29	framework	framework	NOUN
cana-5209	45	30	of	of	ADP
cana-5209	45	31	generalized	generalized	ADJ
cana-5209	45	32	(	(	PUNCT
cana-5209	45	33	c	c	X
cana-5209	45	34	,	,	PUNCT
cana-5209	45	35	α	α	PROPN
cana-5209	45	36	,	,	PUNCT
cana-5209	45	37	η	η	PROPN
cana-5209	45	38	,	,	PUNCT
cana-5209	45	39	ρ	ρ	PROPN
cana-5209	45	40	,	,	PUNCT
cana-5209	45	41	d)-invexity	d)-invexity	NOUN
cana-5209	45	42	.	.	PUNCT
cana-5209	46	1	2	2	NUM
cana-5209	46	2	preliminaries	preliminary	NOUN
cana-5209	46	3	in	in	ADP
cana-5209	46	4	an	an	DET
cana-5209	46	5	n	n	ADV
cana-5209	46	6	-	-	PUNCT
cana-5209	46	7	dimensional	dimensional	ADJ
cana-5209	46	8	euclidean	euclidean	ADJ
cana-5209	46	9	space	space	NOUN
cana-5209	46	10	rn	rn	PROPN
cana-5209	46	11	,	,	PUNCT
cana-5209	46	12	let	let	VERB
cana-5209	46	13	rn	rn	PROPN
cana-5209	46	14	+	+	VERB
cana-5209	46	15	is	be	AUX
cana-5209	46	16	the	the	DET
cana-5209	46	17	non	non	ADJ
cana-5209	46	18	-	-	ADJ
cana-5209	46	19	negative	negative	ADJ
cana-5209	46	20	orthant	orthant	NOUN
cana-5209	46	21	.	.	PUNCT
cana-5209	47	1	consider	consider	VERB
cana-5209	47	2	the	the	DET
cana-5209	47	3	nonlinear	nonlinear	ADJ
cana-5209	47	4	multiobjective	multiobjective	ADJ
cana-5209	47	5	semiinfinite	semiinfinite	ADJ
cana-5209	47	6	programming	programming	NOUN
cana-5209	47	7	problem	problem	NOUN
cana-5209	47	8	defined	define	VERB
cana-5209	47	9	as	as	ADP
cana-5209	47	10	:	:	PUNCT
cana-5209	47	11	(	(	PUNCT
cana-5209	47	12	sp	sp	NOUN
cana-5209	47	13	)	)	PUNCT
cana-5209	47	14	min	min	NOUN
cana-5209	47	15	f(x	f(x	PROPN
cana-5209	47	16	)	)	PUNCT
cana-5209	47	17	subject	subject	NOUN
cana-5209	47	18	to	to	ADP
cana-5209	47	19	gj(x	gj(x	PUNCT
cana-5209	47	20	)	)	PUNCT
cana-5209	47	21	≦	≦	NUM
cana-5209	47	22	0	0	NUM
cana-5209	47	23	,	,	PUNCT
cana-5209	47	24	j	j	PROPN
cana-5209	47	25	∈	∈	PROPN
cana-5209	47	26	j	j	PROPN
cana-5209	47	27	,	,	PUNCT
cana-5209	47	28	let	let	VERB
cana-5209	47	29	f(x	f(x	PROPN
cana-5209	47	30	)	)	PUNCT
cana-5209	48	1	=	=	PRON
cana-5209	48	2	(	(	PUNCT
cana-5209	48	3	f1(x	f1(x	NOUN
cana-5209	48	4	)	)	PUNCT
cana-5209	48	5	,	,	PUNCT
cana-5209	48	6	...	...	PUNCT
cana-5209	48	7	,	,	PUNCT
cana-5209	48	8	fp(x	fp(x	PROPN
cana-5209	48	9	)	)	PUNCT
cana-5209	48	10	)	)	PUNCT
cana-5209	48	11	,	,	PUNCT
cana-5209	48	12	where	where	SCONJ
cana-5209	48	13	each	each	DET
cana-5209	48	14	fi(i	fi(i	X
cana-5209	48	15	∈	∈	PROPN
cana-5209	48	16	p	p	PROPN
cana-5209	48	17	≡	≡	PROPN
cana-5209	48	18	{	{	PUNCT
cana-5209	48	19	1	1	NUM
cana-5209	48	20	,	,	PUNCT
cana-5209	48	21	2	2	NUM
cana-5209	48	22	,	,	PUNCT
cana-5209	48	23	...	...	PUNCT
cana-5209	48	24	,	,	PUNCT
cana-5209	48	25	p	p	X
cana-5209	48	26	}	}	PUNCT
cana-5209	48	27	)	)	PUNCT
cana-5209	48	28	and	and	CCONJ
cana-5209	48	29	gj(j	gj(j	NOUN
cana-5209	48	30	∈	∈	PROPN
cana-5209	48	31	j	j	PROPN
cana-5209	48	32	)	)	PUNCT
cana-5209	48	33	are	be	AUX
cana-5209	48	34	differentiable	differentiable	ADJ
cana-5209	48	35	functions	function	NOUN
cana-5209	48	36	defined	define	VERB
cana-5209	48	37	on	on	ADP
cana-5209	48	38	a	a	DET
cana-5209	48	39	non	non	ADJ
cana-5209	48	40	-	-	ADJ
cana-5209	48	41	empty	empty	ADJ
cana-5209	48	42	open	open	ADJ
cana-5209	48	43	subset	subset	NOUN
cana-5209	48	44	x	x	X
cana-5209	48	45	⊆	⊆	NUM
cana-5209	48	46	rn	rn	NOUN
cana-5209	48	47	and	and	CCONJ
cana-5209	48	48	map	map	VERB
cana-5209	48	49	to	to	PART
cana-5209	48	50	r.	r.	VERB
cana-5209	48	51	the	the	DET
cana-5209	48	52	set	set	PROPN
cana-5209	48	53	j	j	PROPN
cana-5209	48	54	is	be	AUX
cana-5209	48	55	an	an	DET
cana-5209	48	56	index	index	NOUN
cana-5209	48	57	set	set	NOUN
cana-5209	48	58	that	that	PRON
cana-5209	48	59	may	may	AUX
cana-5209	48	60	be	be	AUX
cana-5209	48	61	infinite	infinite	ADJ
cana-5209	48	62	.	.	PUNCT
cana-5209	49	1	here	here	ADV
cana-5209	49	2	we	we	PRON
cana-5209	49	3	take	take	VERB
cana-5209	49	4	the	the	DET
cana-5209	49	5	non	non	ADJ
cana-5209	49	6	-	-	ADJ
cana-5209	49	7	empty	empty	ADJ
cana-5209	49	8	feasible	feasible	ADJ
cana-5209	49	9	set	set	NOUN
cana-5209	49	10	s	s	PRON
cana-5209	49	11	of	of	ADP
cana-5209	49	12	(	(	PUNCT
cana-5209	49	13	sp	sp	NOUN
cana-5209	49	14	):	):	PUNCT
cana-5209	49	15	s	s	PART
cana-5209	49	16	=	=	PUNCT
cana-5209	49	17	{	{	PUNCT
cana-5209	49	18	x	x	SYM
cana-5209	49	19	∈	∈	PROPN
cana-5209	49	20	x	x	X
cana-5209	49	21	:	:	PUNCT
cana-5209	49	22	gj(x	gj(x	X
cana-5209	49	23	)	)	PUNCT
cana-5209	49	24	≦	≦	NUM
cana-5209	49	25	0	0	NUM
cana-5209	49	26	,	,	PUNCT
cana-5209	49	27	j	j	PROPN
cana-5209	49	28	∈	∈	PROPN
cana-5209	49	29	j	j	PROPN
cana-5209	49	30	}	}	PUNCT
cana-5209	49	31	.	.	PUNCT
cana-5209	50	1	and	and	CCONJ
cana-5209	50	2	i	i	PRON
cana-5209	50	3	=	=	PUNCT
cana-5209	50	4	{	{	PUNCT
cana-5209	50	5	j	j	PROPN
cana-5209	50	6	∈	∈	PROPN
cana-5209	50	7	j	j	PROPN
cana-5209	50	8	:	:	PUNCT
cana-5209	50	9	gj(x0	gj(x0	ADJ
cana-5209	50	10	)	)	PUNCT
cana-5209	50	11	=	=	SYM
cana-5209	50	12	0	0	X
cana-5209	50	13	}	}	PUNCT
cana-5209	50	14	where	where	SCONJ
cana-5209	50	15	i	i	PRON
cana-5209	50	16	represents	represent	VERB
cana-5209	50	17	the	the	DET
cana-5209	50	18	index	index	NOUN
cana-5209	50	19	set	set	NOUN
cana-5209	50	20	of	of	ADP
cana-5209	50	21	active	active	ADJ
cana-5209	50	22	constraints	constraint	NOUN
cana-5209	50	23	for	for	ADP
cana-5209	50	24	x0	x0	PROPN
cana-5209	50	25	∈	∈	PROPN
cana-5209	50	26	s.	s.	PROPN
cana-5209	50	27	let	let	VERB
cana-5209	50	28	c	c	NOUN
cana-5209	50	29	:	:	PUNCT
cana-5209	50	30	x×x×rn	x×x×rn	PROPN
cana-5209	51	1	→	→	PUNCT
cana-5209	51	2	r	r	NOUN
cana-5209	51	3	be	be	AUX
cana-5209	51	4	a	a	DET
cana-5209	51	5	function	function	NOUN
cana-5209	51	6	such	such	ADJ
cana-5209	51	7	that	that	PRON
cana-5209	51	8	for	for	ADP
cana-5209	51	9	any	any	DET
cana-5209	51	10	(	(	PUNCT
cana-5209	51	11	x	x	NOUN
cana-5209	51	12	,	,	PUNCT
cana-5209	51	13	x0	x0	PROPN
cana-5209	51	14	)	)	PUNCT
cana-5209	51	15	∈	∈	PROPN
cana-5209	51	16	x×x	x×x	PROPN
cana-5209	51	17	,	,	PUNCT
cana-5209	51	18	it	it	PRON
cana-5209	51	19	satisfies	satisfy	VERB
cana-5209	51	20	c(x	c(x	NOUN
cana-5209	51	21	,	,	PUNCT
cana-5209	51	22	x0)(0	x0)(0	PRON
cana-5209	51	23	)	)	PUNCT
cana-5209	52	1	=	=	SYM
cana-5209	52	2	0	0	X
cana-5209	52	3	.	.	PUNCT
cana-5209	53	1	additionally	additionally	ADV
cana-5209	53	2	,	,	PUNCT
cana-5209	53	3	let	let	VERB
cana-5209	53	4	α	α	PRON
cana-5209	53	5	:	:	PUNCT
cana-5209	53	6	x	x	PROPN
cana-5209	53	7	×x	×x	ADP
cana-5209	53	8	→	→	SYM
cana-5209	53	9	r+	r+	NOUN
cana-5209	53	10	∖	∖	X
cana-5209	53	11	{	{	PUNCT
cana-5209	53	12	0	0	NUM
cana-5209	53	13	}	}	PUNCT
cana-5209	53	14	,	,	PUNCT
cana-5209	53	15	ρ	ρ	PROPN
cana-5209	53	16	∈	∈	PROPN
cana-5209	53	17	r	r	NOUN
cana-5209	53	18	,	,	PUNCT
cana-5209	53	19	and	and	CCONJ
cana-5209	53	20	d	d	NOUN
cana-5209	53	21	:	:	PUNCT
cana-5209	53	22	x	x	X
cana-5209	53	23	×x	×x	ADP
cana-5209	53	24	→	→	SYM
cana-5209	53	25	r+	r+	X
cana-5209	53	26	,	,	PUNCT
cana-5209	53	27	(	(	PUNCT
cana-5209	53	28	d(x	d(x	PROPN
cana-5209	53	29	,	,	PUNCT
cana-5209	53	30	x0	x0	PROPN
cana-5209	53	31	)	)	PUNCT
cana-5209	54	1	=	=	SYM
cana-5209	54	2	0	0	NUM
cana-5209	54	3	iff	iff	NOUN
cana-5209	54	4	x	x	X
cana-5209	54	5	=	=	SYM
cana-5209	54	6	x0	x0	PROPN
cana-5209	54	7	)	)	PUNCT
cana-5209	54	8	.	.	PUNCT
cana-5209	55	1	definition	definition	NOUN
cana-5209	55	2	2.1	2.1	NUM
cana-5209	55	3	.	.	PUNCT
cana-5209	56	1	a	a	DET
cana-5209	56	2	function	function	NOUN
cana-5209	56	3	c	c	NOUN
cana-5209	56	4	:	:	PUNCT
cana-5209	56	5	x	x	X
cana-5209	57	1	×x	×x	PRON
cana-5209	57	2	×rn	×rn	NOUN
cana-5209	57	3	→	→	SYM
cana-5209	57	4	r	r	NOUN
cana-5209	57	5	is	be	AUX
cana-5209	57	6	convex	convex	ADJ
cana-5209	57	7	on	on	ADP
cana-5209	57	8	rn	rn	PROPN
cana-5209	57	9	if	if	SCONJ
cana-5209	58	1	and	and	CCONJ
cana-5209	58	2	only	only	ADV
cana-5209	58	3	if	if	SCONJ
cana-5209	58	4	for	for	ADP
cana-5209	58	5	any	any	DET
cana-5209	58	6	fixed	fixed	ADJ
cana-5209	58	7	point	point	NOUN
cana-5209	58	8	(	(	PUNCT
cana-5209	58	9	x	x	NOUN
cana-5209	58	10	,	,	PUNCT
cana-5209	58	11	x0	x0	PROPN
cana-5209	58	12	)	)	PUNCT
cana-5209	58	13	∈	∈	PROPN
cana-5209	58	14	x	x	X
cana-5209	58	15	×x	×x	X
cana-5209	58	16	and	and	CCONJ
cana-5209	58	17	for	for	ADP
cana-5209	58	18	any	any	DET
cana-5209	58	19	y1	y1	NOUN
cana-5209	58	20	,	,	PUNCT
cana-5209	58	21	y2	y2	PROPN
cana-5209	58	22	∈	∈	PROPN
cana-5209	58	23	rn	rn	PROPN
cana-5209	58	24	,	,	PUNCT
cana-5209	58	25	the	the	DET
cana-5209	58	26	following	follow	VERB
cana-5209	58	27	condition	condition	NOUN
cana-5209	58	28	holds	hold	VERB
cana-5209	58	29	:	:	PUNCT
cana-5209	58	30	c(x	c(x	NOUN
cana-5209	58	31	,	,	PUNCT
cana-5209	58	32	x0)(λy1	x0)(λy1	PUNCT
cana-5209	58	33	+	+	CCONJ
cana-5209	58	34	(	(	PUNCT
cana-5209	58	35	1−	1−	NUM
cana-5209	58	36	λ)y2	λ)y2	PROPN
cana-5209	58	37	)	)	PUNCT
cana-5209	58	38	≦	≦	NUM
cana-5209	58	39	λc(x	λc(x	NOUN
cana-5209	58	40	,	,	PUNCT
cana-5209	58	41	x0)(y1	x0)(y1	PUNCT
cana-5209	58	42	)	)	PUNCT
cana-5209	59	1	+	+	CCONJ
cana-5209	59	2	(	(	PUNCT
cana-5209	59	3	1−	1−	NUM
cana-5209	59	4	λ)c(x	λ)c(x	PROPN
cana-5209	59	5	,	,	PUNCT
cana-5209	59	6	x0)(y2	x0)(y2	NUM
cana-5209	59	7	)	)	PUNCT
cana-5209	59	8	,	,	PUNCT
cana-5209	59	9	∀	∀	PUNCT
cana-5209	59	10	λ	λ	X
cana-5209	59	11	∈	∈	PROPN
cana-5209	59	12	(	(	PUNCT
cana-5209	59	13	0	0	NUM
cana-5209	59	14	,	,	PUNCT
cana-5209	59	15	1	1	NUM
cana-5209	59	16	)	)	PUNCT
cana-5209	59	17	.	.	PUNCT
cana-5209	60	1	definition	definition	NOUN
cana-5209	60	2	2.2	2.2	NUM
cana-5209	60	3	.	.	PUNCT
cana-5209	61	1	a	a	DET
cana-5209	61	2	feasible	feasible	ADJ
cana-5209	61	3	point	point	NOUN
cana-5209	61	4	x0	x0	PROPN
cana-5209	61	5	∈	∈	PROPN
cana-5209	61	6	x	x	PUNCT
cana-5209	61	7	is	be	AUX
cana-5209	61	8	considered	consider	VERB
cana-5209	61	9	an	an	DET
cana-5209	61	10	efficient	efficient	ADJ
cana-5209	61	11	solution	solution	NOUN
cana-5209	61	12	for	for	ADP
cana-5209	61	13	problem	problem	NOUN
cana-5209	61	14	(	(	PUNCT
cana-5209	61	15	sp	sp	NOUN
cana-5209	61	16	)	)	PUNCT
cana-5209	61	17	if	if	SCONJ
cana-5209	62	1	and	and	CCONJ
cana-5209	62	2	only	only	ADV
cana-5209	62	3	if	if	SCONJ
cana-5209	62	4	there	there	PRON
cana-5209	62	5	is	be	VERB
cana-5209	62	6	no	no	DET
cana-5209	62	7	point	point	NOUN
cana-5209	62	8	x	x	X
cana-5209	62	9	∈	∈	NOUN
cana-5209	62	10	x	x	PUNCT
cana-5209	62	11	such	such	ADJ
cana-5209	62	12	that	that	SCONJ
cana-5209	62	13	:	:	PUNCT
cana-5209	62	14	f(x	f(x	PROPN
cana-5209	62	15	)	)	PUNCT
cana-5209	62	16	≤	≤	NUM
cana-5209	62	17	f(x0	f(x0	NOUN
cana-5209	62	18	)	)	PUNCT
cana-5209	62	19	.	.	PUNCT
cana-5209	63	1	definition	definition	NOUN
cana-5209	63	2	2.3	2.3	NUM
cana-5209	63	3	.	.	PUNCT
cana-5209	64	1	a	a	DET
cana-5209	64	2	feasible	feasible	ADJ
cana-5209	64	3	point	point	NOUN
cana-5209	64	4	x0	x0	PROPN
cana-5209	64	5	∈	∈	PROPN
cana-5209	64	6	x	x	X
cana-5209	64	7	is	be	AUX
cana-5209	64	8	a	a	DET
cana-5209	64	9	weak	weak	ADJ
cana-5209	64	10	efficient	efficient	ADJ
cana-5209	64	11	solution	solution	NOUN
cana-5209	64	12	for	for	ADP
cana-5209	64	13	problem	problem	NOUN
cana-5209	64	14	(	(	PUNCT
cana-5209	64	15	sp	sp	NOUN
cana-5209	64	16	)	)	PUNCT
cana-5209	64	17	iff	iff	NOUN
cana-5209	64	18	there	there	PRON
cana-5209	64	19	is	be	VERB
cana-5209	64	20	no	no	DET
cana-5209	64	21	point	point	NOUN
cana-5209	64	22	x	x	X
cana-5209	64	23	∈	∈	NOUN
cana-5209	64	24	x	x	PUNCT
cana-5209	64	25	such	such	ADJ
cana-5209	64	26	that	that	SCONJ
cana-5209	64	27	:	:	PUNCT
cana-5209	64	28	f(x	f(x	PROPN
cana-5209	64	29	)	)	PUNCT
cana-5209	64	30	<	<	X
cana-5209	64	31	f(x0	f(x0	PROPN
cana-5209	64	32	)	)	PUNCT
cana-5209	64	33	.	.	PUNCT
cana-5209	65	1	https://internationalpubls.com	https://internationalpubls.com	X
cana-5209	65	2	1141	1141	NUM
cana-5209	65	3	communications	communication	NOUN
cana-5209	65	4	on	on	ADP
cana-5209	65	5	applied	apply	VERB
cana-5209	65	6	nonlinear	nonlinear	ADJ
cana-5209	65	7	analysis	analysis	NOUN
cana-5209	65	8	issn	issn	NOUN
cana-5209	65	9	:	:	PUNCT
cana-5209	65	10	1074	1074	NUM
cana-5209	65	11	-	-	PUNCT
cana-5209	65	12	133x	133x	NUM
cana-5209	65	13	vol	vol	VERB
cana-5209	65	14	32	32	NUM
cana-5209	65	15	no	no	NOUN
cana-5209	65	16	.	.	PUNCT
cana-5209	66	1	10s	10	NOUN
cana-5209	66	2	(	(	PUNCT
cana-5209	66	3	2025	2025	NUM
cana-5209	66	4	)	)	PUNCT
cana-5209	66	5	the	the	DET
cana-5209	66	6	necessary	necessary	ADJ
cana-5209	66	7	optimality	optimality	NOUN
cana-5209	66	8	conditions	condition	NOUN
cana-5209	66	9	for	for	ADP
cana-5209	66	10	(	(	PUNCT
cana-5209	66	11	sp	sp	NOUN
cana-5209	66	12	)	)	PUNCT
cana-5209	66	13	presented	present	VERB
cana-5209	66	14	below	below	ADV
cana-5209	66	15	are	be	AUX
cana-5209	66	16	taken	take	VERB
cana-5209	66	17	from	from	ADP
cana-5209	66	18	[	[	X
cana-5209	66	19	19	19	NUM
cana-5209	66	20	]	]	PUNCT
cana-5209	66	21	theorem	theorem	VERB
cana-5209	66	22	2.4	2.4	NUM
cana-5209	66	23	.	.	PUNCT
cana-5209	67	1	(	(	PUNCT
cana-5209	67	2	necessary	necessary	ADJ
cana-5209	67	3	optimality	optimality	NOUN
cana-5209	67	4	conditions	condition	NOUN
cana-5209	67	5	)	)	PUNCT
cana-5209	67	6	let	let	VERB
cana-5209	67	7	x0	x0	PROPN
cana-5209	67	8	be	be	AUX
cana-5209	67	9	an	an	DET
cana-5209	67	10	efficient	efficient	ADJ
cana-5209	67	11	solution	solution	NOUN
cana-5209	67	12	for	for	ADP
cana-5209	67	13	(	(	PUNCT
cana-5209	67	14	sp	sp	NOUN
cana-5209	67	15	)	)	PUNCT
cana-5209	67	16	and	and	CCONJ
cana-5209	67	17	i(x0	i(x0	NOUN
cana-5209	67	18	)	)	PUNCT
cana-5209	67	19	̸=	̸=	PROPN
cana-5209	67	20	ϕ.	ϕ.	NOUN
cana-5209	67	21	if	if	SCONJ
cana-5209	67	22	(	(	PUNCT
cana-5209	67	23	sp	sp	NOUN
cana-5209	67	24	)	)	PUNCT
cana-5209	67	25	satisfies	satisfy	VERB
cana-5209	67	26	the	the	DET
cana-5209	67	27	suitable	suitable	ADJ
cana-5209	67	28	constraint	constraint	NOUN
cana-5209	67	29	qualification	qualification	NOUN
cana-5209	67	30	(	(	PUNCT
cana-5209	67	31	see	see	VERB
cana-5209	67	32	[	[	X
cana-5209	67	33	26	26	NUM
cana-5209	67	34	]	]	PUNCT
cana-5209	67	35	)	)	PUNCT
cana-5209	67	36	at	at	ADP
cana-5209	67	37	x0	x0	PROPN
cana-5209	67	38	then	then	ADV
cana-5209	67	39	there	there	PRON
cana-5209	67	40	exist	exist	VERB
cana-5209	67	41	u	u	PROPN
cana-5209	67	42	∈	∈	PROPN
cana-5209	67	43	rp	rp	NOUN
cana-5209	67	44	,	,	PUNCT
cana-5209	67	45	v	v	NOUN
cana-5209	67	46	=	=	SYM
cana-5209	67	47	(	(	PUNCT
cana-5209	67	48	vj)j∈j	vj)j∈j	NUM
cana-5209	67	49	,	,	PUNCT
cana-5209	67	50	such	such	ADJ
cana-5209	67	51	that	that	SCONJ
cana-5209	67	52	ut∇f(x0	ut∇f(x0	NOUN
cana-5209	67	53	)	)	PUNCT
cana-5209	67	54	+	+	CCONJ
cana-5209	67	55	vi	vi	X
cana-5209	67	56	t∇gi(x0	t∇gi(x0	NOUN
cana-5209	67	57	)	)	PUNCT
cana-5209	68	1	=	=	SYM
cana-5209	68	2	0	0	NUM
cana-5209	68	3	,	,	PUNCT
cana-5209	68	4	(	(	PUNCT
cana-5209	68	5	2.1	2.1	NUM
cana-5209	68	6	)	)	PUNCT
cana-5209	68	7	vt	vt	NOUN
cana-5209	68	8	g(x0	g(x0	NOUN
cana-5209	68	9	)	)	PUNCT
cana-5209	69	1	=	=	SYM
cana-5209	69	2	0	0	NUM
cana-5209	69	3	,	,	PUNCT
cana-5209	69	4	u	u	NOUN
cana-5209	69	5	≥	≥	NOUN
cana-5209	69	6	0	0	NUM
cana-5209	69	7	,	,	PUNCT
cana-5209	69	8	v	v	PRON
cana-5209	69	9	≥	≥	NOUN
cana-5209	69	10	0	0	NUM
cana-5209	69	11	and	and	CCONJ
cana-5209	69	12	vj	vj	INTJ
cana-5209	69	13	̸=	̸=	PROPN
cana-5209	69	14	0	0	NUM
cana-5209	69	15	for	for	ADP
cana-5209	69	16	finitely	finitely	ADV
cana-5209	69	17	many	many	ADJ
cana-5209	69	18	j	j	PROPN
cana-5209	69	19	∈	∈	PROPN
cana-5209	69	20	i.	i.	NOUN
cana-5209	69	21	3	3	NUM
cana-5209	69	22	definitions	definition	NOUN
cana-5209	69	23	in	in	ADP
cana-5209	69	24	2006	2006	NUM
cana-5209	69	25	,	,	PUNCT
cana-5209	69	26	yuan	yuan	NOUN
cana-5209	69	27	and	and	CCONJ
cana-5209	69	28	liu	liu	PROPN
cana-5209	69	29	(	(	PUNCT
cana-5209	69	30	see	see	VERB
cana-5209	69	31	[	[	X
cana-5209	69	32	13	13	NUM
cana-5209	69	33	]	]	PUNCT
cana-5209	69	34	)	)	PUNCT
cana-5209	69	35	gave	give	VERB
cana-5209	69	36	the	the	DET
cana-5209	69	37	definition	definition	NOUN
cana-5209	69	38	of	of	ADP
cana-5209	69	39	(	(	PUNCT
cana-5209	69	40	c	c	X
cana-5209	69	41	,	,	PUNCT
cana-5209	69	42	α	α	PROPN
cana-5209	69	43	,	,	PUNCT
cana-5209	69	44	ρ	ρ	PROPN
cana-5209	69	45	,	,	PUNCT
cana-5209	69	46	d	d	NOUN
cana-5209	69	47	)	)	PUNCT
cana-5209	69	48	convex	convex	NOUN
cana-5209	69	49	function	function	NOUN
cana-5209	69	50	.	.	PUNCT
cana-5209	70	1	we	we	PRON
cana-5209	70	2	see	see	VERB
cana-5209	70	3	that	that	SCONJ
cana-5209	70	4	this	this	DET
cana-5209	70	5	definition	definition	NOUN
cana-5209	70	6	is	be	AUX
cana-5209	70	7	not	not	PART
cana-5209	70	8	hold	hold	VERB
cana-5209	70	9	when	when	SCONJ
cana-5209	70	10	α	α	PRON
cana-5209	70	11	≤	≤	NOUN
cana-5209	70	12	0	0	NUM
cana-5209	70	13	.	.	PUNCT
cana-5209	71	1	in	in	ADP
cana-5209	71	2	this	this	DET
cana-5209	71	3	paper	paper	NOUN
cana-5209	71	4	,	,	PUNCT
cana-5209	71	5	we	we	PRON
cana-5209	71	6	introduce	introduce	VERB
cana-5209	71	7	the	the	DET
cana-5209	71	8	definitions	definition	NOUN
cana-5209	71	9	of	of	ADP
cana-5209	71	10	(	(	PUNCT
cana-5209	71	11	c	c	X
cana-5209	71	12	,	,	PUNCT
cana-5209	71	13	α	α	PROPN
cana-5209	71	14	,	,	PUNCT
cana-5209	71	15	η	η	PROPN
cana-5209	71	16	,	,	PUNCT
cana-5209	71	17	ρ	ρ	PROPN
cana-5209	71	18	,	,	PUNCT
cana-5209	71	19	d	d	NOUN
cana-5209	71	20	)	)	PUNCT
cana-5209	71	21	invex	invex	NOUN
cana-5209	71	22	function	function	NOUN
cana-5209	71	23	and	and	CCONJ
cana-5209	71	24	its	its	PRON
cana-5209	71	25	generalizations	generalization	NOUN
cana-5209	71	26	,	,	PUNCT
cana-5209	71	27	which	which	PRON
cana-5209	71	28	will	will	AUX
cana-5209	71	29	also	also	ADV
cana-5209	71	30	hold	hold	VERB
cana-5209	71	31	when	when	SCONJ
cana-5209	71	32	η	η	PROPN
cana-5209	71	33	≤	≤	X
cana-5209	71	34	0	0	NUM
cana-5209	71	35	.	.	PUNCT
cana-5209	72	1	these	these	DET
cana-5209	72	2	definitions	definition	NOUN
cana-5209	72	3	will	will	AUX
cana-5209	72	4	be	be	AUX
cana-5209	72	5	used	use	VERB
cana-5209	72	6	in	in	ADP
cana-5209	72	7	the	the	DET
cana-5209	72	8	sequal	sequal	ADJ
cana-5209	72	9	.	.	PUNCT
cana-5209	73	1	definition	definition	NOUN
cana-5209	73	2	3.1	3.1	NUM
cana-5209	73	3	.	.	PUNCT
cana-5209	74	1	the	the	DET
cana-5209	74	2	vector	vector	NOUN
cana-5209	74	3	-	-	PUNCT
cana-5209	74	4	valued	value	VERB
cana-5209	74	5	function	function	NOUN
cana-5209	74	6	f	f	NOUN
cana-5209	74	7	:	:	PUNCT
cana-5209	74	8	x	x	X
cana-5209	74	9	→	→	SYM
cana-5209	74	10	rp	rp	NOUN
cana-5209	74	11	is	be	AUX
cana-5209	74	12	called	call	VERB
cana-5209	74	13	(	(	PUNCT
cana-5209	74	14	c	c	PROPN
cana-5209	74	15	,	,	PUNCT
cana-5209	74	16	α	α	PROPN
cana-5209	74	17	,	,	PUNCT
cana-5209	74	18	η	η	PROPN
cana-5209	74	19	,	,	PUNCT
cana-5209	74	20	ρ	ρ	PROPN
cana-5209	74	21	,	,	PUNCT
cana-5209	74	22	d)-invex	d)-invex	X
cana-5209	74	23	at	at	ADP
cana-5209	74	24	x0	x0	PROPN
cana-5209	74	25	∈	∈	PROPN
cana-5209	74	26	x	x	INTJ
cana-5209	74	27	if	if	SCONJ
cana-5209	74	28	,	,	PUNCT
cana-5209	74	29	for	for	ADP
cana-5209	74	30	each	each	DET
cana-5209	74	31	fi	fi	NOUN
cana-5209	74	32	:	:	PUNCT
cana-5209	74	33	x	x	X
cana-5209	74	34	→	→	SYM
cana-5209	74	35	r	r	NOUN
cana-5209	74	36	,	,	PUNCT
cana-5209	74	37	there	there	PRON
cana-5209	74	38	exists	exist	VERB
cana-5209	74	39	a	a	DET
cana-5209	74	40	vectorial	vectorial	ADJ
cana-5209	74	41	function	function	NOUN
cana-5209	74	42	ηi	ηi	NOUN
cana-5209	74	43	:	:	PUNCT
cana-5209	74	44	x	x	PUNCT
cana-5209	74	45	×	×	NOUN
cana-5209	74	46	x	x	INTJ
cana-5209	74	47	→	→	PUNCT
cana-5209	74	48	r	r	NOUN
cana-5209	74	49	such	such	ADJ
cana-5209	74	50	that	that	PRON
cana-5209	74	51	for	for	ADP
cana-5209	74	52	every	every	DET
cana-5209	74	53	x	x	SYM
cana-5209	74	54	∈	∈	PROPN
cana-5209	74	55	x	x	X
cana-5209	74	56	and	and	CCONJ
cana-5209	74	57	i	i	PRON
cana-5209	74	58	∈	∈	PROPN
cana-5209	74	59	p	p	X
cana-5209	74	60	,	,	PUNCT
cana-5209	74	61	the	the	DET
cana-5209	74	62	following	follow	VERB
cana-5209	74	63	condition	condition	NOUN
cana-5209	74	64	holds	hold	VERB
cana-5209	74	65	:	:	PUNCT
cana-5209	74	66	fi(x)−	fi(x)−	PROPN
cana-5209	74	67	fi(x0	fi(x0	PROPN
cana-5209	74	68	)	)	PUNCT
cana-5209	74	69	αi(x	αi(x	NUM
cana-5209	74	70	,	,	PUNCT
cana-5209	74	71	x0	x0	PROPN
cana-5209	74	72	)	)	PUNCT
cana-5209	74	73	(	(	PUNCT
cana-5209	74	74	>	>	PUNCT
cana-5209	74	75	)	)	PUNCT
cana-5209	74	76	≧	≧	X
cana-5209	75	1	c(x	c(x	NOUN
cana-5209	75	2	,	,	PUNCT
cana-5209	75	3	x0)(∇fi(x0))ηi(x	x0)(∇fi(x0))ηi(x	NUM
cana-5209	75	4	,	,	PUNCT
cana-5209	75	5	x0	x0	PROPN
cana-5209	75	6	)	)	PUNCT
cana-5209	76	1	+	+	CCONJ
cana-5209	76	2	ρi	ρi	NOUN
cana-5209	76	3	di(x	di(x	NOUN
cana-5209	76	4	,	,	PUNCT
cana-5209	76	5	x0	x0	PROPN
cana-5209	76	6	)	)	PUNCT
cana-5209	76	7	αi(x	αi(x	NUM
cana-5209	76	8	,	,	PUNCT
cana-5209	76	9	x0	x0	PROPN
cana-5209	76	10	)	)	PUNCT
cana-5209	76	11	.	.	PUNCT
cana-5209	77	1	the	the	DET
cana-5209	77	2	function	function	NOUN
cana-5209	77	3	f	f	PROPN
cana-5209	77	4	is	be	AUX
cana-5209	77	5	said	say	VERB
cana-5209	77	6	to	to	PART
cana-5209	77	7	be	be	AUX
cana-5209	77	8	(	(	PUNCT
cana-5209	77	9	c	c	X
cana-5209	77	10	,	,	PUNCT
cana-5209	77	11	α	α	PROPN
cana-5209	77	12	,	,	PUNCT
cana-5209	77	13	η	η	PROPN
cana-5209	77	14	,	,	PUNCT
cana-5209	77	15	ρ	ρ	PROPN
cana-5209	77	16	,	,	PUNCT
cana-5209	77	17	d)-invex	d)-invex	VERB
cana-5209	77	18	on	on	ADP
cana-5209	77	19	x	x	SYM
cana-5209	77	20	if	if	SCONJ
cana-5209	78	1	and	and	CCONJ
cana-5209	78	2	only	only	ADV
cana-5209	78	3	if	if	SCONJ
cana-5209	78	4	it	it	PRON
cana-5209	78	5	is	be	AUX
cana-5209	78	6	(	(	PUNCT
cana-5209	78	7	c	c	X
cana-5209	78	8	,	,	PUNCT
cana-5209	78	9	α	α	PROPN
cana-5209	78	10	,	,	PUNCT
cana-5209	78	11	η	η	PROPN
cana-5209	78	12	,	,	PUNCT
cana-5209	78	13	ρ	ρ	PROPN
cana-5209	78	14	,	,	PUNCT
cana-5209	78	15	d)-invex	d)-invex	X
cana-5209	78	16	at	at	ADP
cana-5209	78	17	every	every	DET
cana-5209	78	18	point	point	NOUN
cana-5209	78	19	in	in	ADP
cana-5209	78	20	x.	x.	NOUN
cana-5209	78	21	by	by	ADP
cana-5209	78	22	replacing	replace	VERB
cana-5209	78	23	η(x	η(x	NOUN
cana-5209	78	24	,	,	PUNCT
cana-5209	78	25	x0	x0	PROPN
cana-5209	78	26	)	)	PUNCT
cana-5209	78	27	with	with	ADP
cana-5209	78	28	1	1	NUM
cana-5209	78	29	in	in	ADP
cana-5209	78	30	the	the	DET
cana-5209	78	31	above	above	ADJ
cana-5209	78	32	definition	definition	NOUN
cana-5209	78	33	,	,	PUNCT
cana-5209	78	34	we	we	PRON
cana-5209	78	35	will	will	AUX
cana-5209	78	36	get	get	VERB
cana-5209	78	37	the	the	DET
cana-5209	78	38	definition	definition	NOUN
cana-5209	78	39	of	of	ADP
cana-5209	78	40	(	(	PUNCT
cana-5209	78	41	c	c	X
cana-5209	78	42	,	,	PUNCT
cana-5209	78	43	α	α	PROPN
cana-5209	78	44	,	,	PUNCT
cana-5209	78	45	ρ	ρ	PROPN
cana-5209	78	46	,	,	PUNCT
cana-5209	78	47	d	d	NOUN
cana-5209	78	48	)	)	PUNCT
cana-5209	78	49	convexity	convexity	NOUN
cana-5209	78	50	.	.	PUNCT
cana-5209	79	1	in	in	ADP
cana-5209	79	2	this	this	DET
cana-5209	79	3	example	example	NOUN
cana-5209	79	4	,	,	PUNCT
cana-5209	79	5	we	we	PRON
cana-5209	79	6	try	try	VERB
cana-5209	79	7	to	to	PART
cana-5209	79	8	illustrate	illustrate	VERB
cana-5209	79	9	the	the	DET
cana-5209	79	10	importance	importance	NOUN
cana-5209	79	11	of	of	ADP
cana-5209	79	12	invex	invex	NOUN
cana-5209	79	13	functions	function	NOUN
cana-5209	79	14	and	and	CCONJ
cana-5209	79	15	its	its	PRON
cana-5209	79	16	generalization	generalization	NOUN
cana-5209	79	17	.	.	PUNCT
cana-5209	80	1	exmple	exmple	PROPN
cana-5209	80	2	.	.	PUNCT
cana-5209	81	1	let	let	VERB
cana-5209	81	2	x	x	PUNCT
cana-5209	81	3	=	=	PRON
cana-5209	81	4	{	{	PUNCT
cana-5209	81	5	x	x	X
cana-5209	81	6	:	:	PUNCT
cana-5209	81	7	π	π	PROPN
cana-5209	81	8	4	4	NUM
cana-5209	81	9	≤	≤	NUM
cana-5209	81	10	x	x	PUNCT
cana-5209	81	11	≤	≤	NUM
cana-5209	81	12	π	π	X
cana-5209	81	13	2	2	NUM
cana-5209	81	14	}	}	PUNCT
cana-5209	81	15	,	,	PUNCT
cana-5209	81	16	ρ	ρ	PROPN
cana-5209	81	17	=	=	SYM
cana-5209	81	18	−1	−1	NOUN
cana-5209	81	19	,	,	PUNCT
cana-5209	81	20	α(x	α(x	PROPN
cana-5209	81	21	,	,	PUNCT
cana-5209	81	22	x0	x0	PROPN
cana-5209	81	23	)	)	PUNCT
cana-5209	82	1	=	=	SYM
cana-5209	82	2	1	1	NUM
cana-5209	82	3	,	,	PUNCT
cana-5209	82	4	d(x	d(x	PROPN
cana-5209	82	5	,	,	PUNCT
cana-5209	82	6	x0	x0	PROPN
cana-5209	82	7	)	)	PUNCT
cana-5209	83	1	=	=	PRON
cana-5209	83	2	(	(	PUNCT
cana-5209	83	3	x	x	X
cana-5209	83	4	−	−	NOUN
cana-5209	83	5	x0	x0	PROPN
cana-5209	83	6	)	)	PUNCT
cana-5209	83	7	,	,	PUNCT
cana-5209	83	8	c(x	c(x	NOUN
cana-5209	83	9	,	,	PUNCT
cana-5209	83	10	x0)(a	x0)(a	NUM
cana-5209	83	11	)	)	PUNCT
cana-5209	83	12	=	=	SYM
cana-5209	83	13	a2(x−	a2(x−	X
cana-5209	83	14	x0	x0	PROPN
cana-5209	83	15	)	)	PUNCT
cana-5209	83	16	for	for	ADP
cana-5209	83	17	any	any	DET
cana-5209	83	18	(	(	PUNCT
cana-5209	83	19	x	x	NOUN
cana-5209	83	20	,	,	PUNCT
cana-5209	83	21	x0	x0	PROPN
cana-5209	83	22	)	)	PUNCT
cana-5209	83	23	∈	∈	PROPN
cana-5209	83	24	x	x	X
cana-5209	83	25	×x	×x	VERB
cana-5209	83	26	and	and	CCONJ
cana-5209	83	27	let	let	VERB
cana-5209	83	28	f(x	f(x	PROPN
cana-5209	83	29	)	)	PUNCT
cana-5209	83	30	=	=	PUNCT
cana-5209	84	1	cos2x	cos2x	PROPN
cana-5209	84	2	.	.	PUNCT
cana-5209	85	1	then	then	ADV
cana-5209	85	2	,	,	PUNCT
cana-5209	85	3	we	we	PRON
cana-5209	85	4	see	see	VERB
cana-5209	85	5	that	that	SCONJ
cana-5209	85	6	f(x	f(x	PROPN
cana-5209	85	7	)	)	PUNCT
cana-5209	85	8	is	be	AUX
cana-5209	85	9	not	not	PART
cana-5209	85	10	(	(	PUNCT
cana-5209	85	11	c	c	X
cana-5209	85	12	,	,	PUNCT
cana-5209	85	13	α	α	PROPN
cana-5209	85	14	,	,	PUNCT
cana-5209	85	15	ρ	ρ	PROPN
cana-5209	85	16	,	,	PUNCT
cana-5209	85	17	d)convex	d)convex	NOUN
cana-5209	85	18	at	at	ADP
cana-5209	85	19	x	x	X
cana-5209	85	20	=	=	SYM
cana-5209	85	21	π	π	PROPN
cana-5209	85	22	4	4	NUM
cana-5209	85	23	,	,	PUNCT
cana-5209	85	24	but	but	CCONJ
cana-5209	85	25	it	it	PRON
cana-5209	85	26	is	be	AUX
cana-5209	85	27	(	(	PUNCT
cana-5209	85	28	c	c	X
cana-5209	85	29	,	,	PUNCT
cana-5209	85	30	α	α	PROPN
cana-5209	85	31	,	,	PUNCT
cana-5209	85	32	η	η	PROPN
cana-5209	85	33	,	,	PUNCT
cana-5209	85	34	ρ	ρ	PROPN
cana-5209	85	35	,	,	PUNCT
cana-5209	85	36	d)-invex	d)-invex	X
cana-5209	85	37	at	at	ADP
cana-5209	85	38	x	x	X
cana-5209	85	39	=	=	SYM
cana-5209	85	40	π	π	PROPN
cana-5209	85	41	4	4	NUM
cana-5209	85	42	with	with	ADP
cana-5209	85	43	η(x	η(x	NOUN
cana-5209	85	44	,	,	PUNCT
cana-5209	85	45	x0	x0	PROPN
cana-5209	85	46	)	)	PUNCT
cana-5209	86	1	=	=	SYM
cana-5209	86	2	(	(	PUNCT
cana-5209	86	3	1−	1−	NUM
cana-5209	86	4	cos(x−	cos(x−	SYM
cana-5209	86	5	x0	x0	PROPN
cana-5209	86	6	)	)	PUNCT
cana-5209	86	7	(	(	PUNCT
cana-5209	86	8	x−	x−	PROPN
cana-5209	86	9	x0	x0	PROPN
cana-5209	86	10	)	)	PUNCT
cana-5209	86	11	)	)	PUNCT
cana-5209	86	12	.	.	PUNCT
cana-5209	87	1	as	as	SCONJ
cana-5209	87	2	we	we	PRON
cana-5209	87	3	know	know	VERB
cana-5209	87	4	that	that	SCONJ
cana-5209	87	5	the	the	DET
cana-5209	87	6	definition	definition	NOUN
cana-5209	87	7	of	of	ADP
cana-5209	87	8	(	(	PUNCT
cana-5209	87	9	c	c	X
cana-5209	87	10	,	,	PUNCT
cana-5209	87	11	α	α	PROPN
cana-5209	87	12	,	,	PUNCT
cana-5209	87	13	ρ	ρ	PROPN
cana-5209	87	14	,	,	PUNCT
cana-5209	87	15	d	d	NOUN
cana-5209	87	16	)	)	PUNCT
cana-5209	87	17	convexity	convexity	NOUN
cana-5209	87	18	is	be	AUX
cana-5209	87	19	not	not	PART
cana-5209	87	20	hold	hold	ADJ
cana-5209	87	21	when	when	SCONJ
cana-5209	87	22	α	α	PRON
cana-5209	87	23	≤	≤	NOUN
cana-5209	87	24	0	0	NUM
cana-5209	87	25	.	.	PUNCT
cana-5209	88	1	but	but	CCONJ
cana-5209	88	2	on	on	ADP
cana-5209	88	3	the	the	DET
cana-5209	88	4	other	other	ADJ
cana-5209	88	5	hand	hand	NOUN
cana-5209	88	6	we	we	PRON
cana-5209	88	7	see	see	VERB
cana-5209	88	8	that	that	SCONJ
cana-5209	88	9	,	,	PUNCT
cana-5209	88	10	the	the	DET
cana-5209	88	11	definition	definition	NOUN
cana-5209	88	12	of	of	ADP
cana-5209	88	13	(	(	PUNCT
cana-5209	88	14	c	c	X
cana-5209	88	15	,	,	PUNCT
cana-5209	88	16	α	α	PROPN
cana-5209	88	17	,	,	PUNCT
cana-5209	88	18	η	η	PROPN
cana-5209	88	19	,	,	PUNCT
cana-5209	88	20	ρ	ρ	PROPN
cana-5209	88	21	,	,	PUNCT
cana-5209	88	22	d	d	NOUN
cana-5209	88	23	)	)	PUNCT
cana-5209	88	24	invexity	invexity	NOUN
cana-5209	88	25	holds	hold	VERB
cana-5209	88	26	for	for	ADP
cana-5209	88	27	all	all	DET
cana-5209	88	28	real	real	ADJ
cana-5209	88	29	values	value	NOUN
cana-5209	88	30	of	of	ADP
cana-5209	88	31	η	η	PROPN
cana-5209	88	32	.	.	PROPN
cana-5209	88	33	definition	definition	NOUN
cana-5209	88	34	3.2	3.2	NUM
cana-5209	88	35	.	.	PUNCT
cana-5209	89	1	the	the	DET
cana-5209	89	2	vector	vector	NOUN
cana-5209	89	3	-	-	PUNCT
cana-5209	89	4	valued	value	VERB
cana-5209	89	5	function	function	NOUN
cana-5209	89	6	f	f	NOUN
cana-5209	89	7	:	:	PUNCT
cana-5209	89	8	x	x	X
cana-5209	89	9	→	→	SYM
cana-5209	89	10	rp	rp	NOUN
cana-5209	89	11	is	be	AUX
cana-5209	89	12	called	call	VERB
cana-5209	89	13	(	(	PUNCT
cana-5209	89	14	c	c	PROPN
cana-5209	89	15	,	,	PUNCT
cana-5209	89	16	α	α	PROPN
cana-5209	89	17	,	,	PUNCT
cana-5209	89	18	η	η	PROPN
cana-5209	89	19	,	,	PUNCT
cana-5209	89	20	ρ	ρ	NOUN
cana-5209	89	21	,	,	PUNCT
cana-5209	89	22	d)-pseudo	d)-pseudo	NOUN
cana-5209	89	23	-	-	NOUN
cana-5209	89	24	invex	invex	NOUN
cana-5209	89	25	at	at	ADP
cana-5209	89	26	x0	x0	PROPN
cana-5209	89	27	∈	∈	PROPN
cana-5209	89	28	x	x	INTJ
cana-5209	89	29	if	if	SCONJ
cana-5209	89	30	,	,	PUNCT
cana-5209	89	31	for	for	ADP
cana-5209	89	32	each	each	DET
cana-5209	89	33	fi	fi	NOUN
cana-5209	89	34	:	:	PUNCT
cana-5209	89	35	x	x	X
cana-5209	89	36	→	→	SYM
cana-5209	89	37	r	r	NOUN
cana-5209	89	38	,	,	PUNCT
cana-5209	89	39	there	there	PRON
cana-5209	89	40	exists	exist	VERB
cana-5209	89	41	a	a	DET
cana-5209	89	42	vectorial	vectorial	ADJ
cana-5209	89	43	function	function	NOUN
cana-5209	89	44	ηi	ηi	NOUN
cana-5209	89	45	:	:	PUNCT
cana-5209	89	46	x	x	X
cana-5209	89	47	×x	×x	ADP
cana-5209	89	48	→	→	SYM
cana-5209	89	49	r	r	AUX
cana-5209	89	50	such	such	ADJ
cana-5209	89	51	that	that	PRON
cana-5209	89	52	for	for	ADP
cana-5209	89	53	every	every	DET
cana-5209	89	54	x	x	SYM
cana-5209	89	55	∈	∈	PROPN
cana-5209	89	56	x	x	X
cana-5209	89	57	and	and	CCONJ
cana-5209	89	58	i	i	PRON
cana-5209	89	59	∈	∈	PROPN
cana-5209	89	60	p	p	X
cana-5209	89	61	,	,	PUNCT
cana-5209	89	62	the	the	DET
cana-5209	89	63	following	follow	VERB
cana-5209	89	64	condition	condition	NOUN
cana-5209	89	65	holds	hold	VERB
cana-5209	89	66	:	:	PUNCT
cana-5209	89	67	fi(x)(≦	fi(x)(≦	X
cana-5209	89	68	)	)	PUNCT
cana-5209	89	69	<	<	X
cana-5209	89	70	fi(x0	fi(x0	PROPN
cana-5209	89	71	)	)	PUNCT
cana-5209	89	72	⇒	⇒	PROPN
cana-5209	89	73	c(x	c(x	PROPN
cana-5209	89	74	,	,	PUNCT
cana-5209	89	75	x0)(∇fi(x0))ηi(x	x0)(∇fi(x0))ηi(x	NUM
cana-5209	89	76	,	,	PUNCT
cana-5209	89	77	x0	x0	PROPN
cana-5209	89	78	)	)	PUNCT
cana-5209	90	1	+	+	CCONJ
cana-5209	90	2	ρi	ρi	NOUN
cana-5209	90	3	di(x	di(x	NOUN
cana-5209	90	4	,	,	PUNCT
cana-5209	90	5	x0	x0	PROPN
cana-5209	90	6	)	)	PUNCT
cana-5209	90	7	αi(x	αi(x	NUM
cana-5209	90	8	,	,	PUNCT
cana-5209	90	9	x0	x0	PROPN
cana-5209	90	10	)	)	PUNCT
cana-5209	90	11	<	<	X
cana-5209	90	12	0	0	X
cana-5209	90	13	.	.	PUNCT
cana-5209	91	1	the	the	DET
cana-5209	91	2	function	function	NOUN
cana-5209	91	3	f	f	PROPN
cana-5209	91	4	is	be	AUX
cana-5209	91	5	said	say	VERB
cana-5209	91	6	to	to	PART
cana-5209	91	7	be	be	AUX
cana-5209	91	8	(	(	PUNCT
cana-5209	91	9	c	c	X
cana-5209	91	10	,	,	PUNCT
cana-5209	91	11	α	α	PROPN
cana-5209	91	12	,	,	PUNCT
cana-5209	91	13	η	η	PROPN
cana-5209	91	14	,	,	PUNCT
cana-5209	91	15	ρ	ρ	NOUN
cana-5209	91	16	,	,	PUNCT
cana-5209	91	17	d)-pseudo	d)-pseudo	NOUN
cana-5209	91	18	-	-	NOUN
cana-5209	91	19	invex	invex	NOUN
cana-5209	91	20	on	on	ADP
cana-5209	91	21	x	x	PUNCT
cana-5209	91	22	iff	iff	PROPN
cana-5209	91	23	it	it	PRON
cana-5209	91	24	is	be	AUX
cana-5209	91	25	(	(	PUNCT
cana-5209	91	26	c	c	X
cana-5209	91	27	,	,	PUNCT
cana-5209	91	28	α	α	PROPN
cana-5209	91	29	,	,	PUNCT
cana-5209	91	30	η	η	PROPN
cana-5209	91	31	,	,	PUNCT
cana-5209	91	32	ρ	ρ	NOUN
cana-5209	91	33	,	,	PUNCT
cana-5209	91	34	d)-pseudo	d)-pseudo	NOUN
cana-5209	91	35	-	-	NOUN
cana-5209	91	36	invex	invex	NOUN
cana-5209	91	37	at	at	ADP
cana-5209	91	38	each	each	DET
cana-5209	91	39	point	point	NOUN
cana-5209	91	40	in	in	ADP
cana-5209	91	41	x.	x.	NOUN
cana-5209	91	42	definition	definition	NOUN
cana-5209	91	43	3.3	3.3	NUM
cana-5209	91	44	.	.	PUNCT
cana-5209	92	1	the	the	DET
cana-5209	92	2	vector	vector	NOUN
cana-5209	92	3	-	-	PUNCT
cana-5209	92	4	valued	value	VERB
cana-5209	92	5	function	function	NOUN
cana-5209	92	6	f	f	NOUN
cana-5209	92	7	:	:	PUNCT
cana-5209	92	8	x	x	X
cana-5209	92	9	→	→	SYM
cana-5209	92	10	rp	rp	NOUN
cana-5209	92	11	is	be	AUX
cana-5209	92	12	called	call	VERB
cana-5209	92	13	weak	weak	ADJ
cana-5209	92	14	strictly	strictly	ADV
cana-5209	92	15	(	(	PUNCT
cana-5209	92	16	c	c	X
cana-5209	92	17	,	,	PUNCT
cana-5209	92	18	α	α	PROPN
cana-5209	92	19	,	,	PUNCT
cana-5209	92	20	η	η	PROPN
cana-5209	92	21	,	,	PUNCT
cana-5209	92	22	ρ	ρ	PROPN
cana-5209	92	23	,	,	PUNCT
cana-5209	92	24	d)-pseudoinvex	d)-pseudoinvex	PUNCT
cana-5209	92	25	at	at	ADP
cana-5209	92	26	x0	x0	PROPN
cana-5209	92	27	∈	∈	PROPN
cana-5209	92	28	x	x	INTJ
cana-5209	92	29	if	if	SCONJ
cana-5209	92	30	,	,	PUNCT
cana-5209	92	31	for	for	ADP
cana-5209	92	32	each	each	DET
cana-5209	92	33	fi	fi	NOUN
cana-5209	92	34	:	:	PUNCT
cana-5209	92	35	x	x	X
cana-5209	92	36	→	→	SYM
cana-5209	92	37	r	r	NOUN
cana-5209	92	38	there	there	PRON
cana-5209	92	39	exists	exist	VERB
cana-5209	92	40	a	a	DET
cana-5209	92	41	vectorial	vectorial	ADJ
cana-5209	92	42	function	function	NOUN
cana-5209	92	43	ηi	ηi	NOUN
cana-5209	92	44	:	:	PUNCT
cana-5209	92	45	x	x	X
cana-5209	92	46	×x	×x	ADP
cana-5209	92	47	→	→	SYM
cana-5209	92	48	r	r	NOUN
cana-5209	92	49	such	such	ADJ
cana-5209	92	50	that	that	PRON
cana-5209	92	51	for	for	ADP
cana-5209	92	52	every	every	DET
cana-5209	92	53	x	x	SYM
cana-5209	92	54	∈	∈	PROPN
cana-5209	92	55	x	x	X
cana-5209	92	56	and	and	CCONJ
cana-5209	92	57	i	i	PRON
cana-5209	92	58	∈	∈	PROPN
cana-5209	92	59	p	p	X
cana-5209	92	60	,	,	PUNCT
cana-5209	92	61	the	the	DET
cana-5209	92	62	following	follow	VERB
cana-5209	92	63	condition	condition	NOUN
cana-5209	92	64	holds	hold	VERB
cana-5209	92	65	:	:	PUNCT
cana-5209	92	66	fi(x	fi(x	NUM
cana-5209	92	67	)	)	PUNCT
cana-5209	92	68	≤	≤	NUM
cana-5209	92	69	fi(x0	fi(x0	NOUN
cana-5209	92	70	)	)	PUNCT
cana-5209	92	71	⇒	⇒	PROPN
cana-5209	92	72	c(x	c(x	PROPN
cana-5209	92	73	,	,	PUNCT
cana-5209	92	74	x0)(∇fi(x0))ηi(x	x0)(∇fi(x0))ηi(x	NUM
cana-5209	92	75	,	,	PUNCT
cana-5209	92	76	x0	x0	PROPN
cana-5209	92	77	)	)	PUNCT
cana-5209	93	1	+	+	CCONJ
cana-5209	93	2	ρi	ρi	NOUN
cana-5209	93	3	di(x	di(x	NOUN
cana-5209	93	4	,	,	PUNCT
cana-5209	93	5	x0	x0	PROPN
cana-5209	93	6	)	)	PUNCT
cana-5209	93	7	αi(x	αi(x	NUM
cana-5209	93	8	,	,	PUNCT
cana-5209	93	9	x0	x0	PROPN
cana-5209	93	10	)	)	PUNCT
cana-5209	93	11	<	<	X
cana-5209	93	12	0	0	X
cana-5209	93	13	.	.	PUNCT
cana-5209	94	1	the	the	DET
cana-5209	94	2	function	function	NOUN
cana-5209	94	3	f	f	PROPN
cana-5209	94	4	is	be	AUX
cana-5209	94	5	said	say	VERB
cana-5209	94	6	to	to	PART
cana-5209	94	7	be	be	AUX
cana-5209	94	8	weak	weak	ADJ
cana-5209	94	9	strictly	strictly	ADV
cana-5209	94	10	(	(	PUNCT
cana-5209	94	11	c	c	X
cana-5209	94	12	,	,	PUNCT
cana-5209	94	13	α	α	PROPN
cana-5209	94	14	,	,	PUNCT
cana-5209	94	15	η	η	PROPN
cana-5209	94	16	,	,	PUNCT
cana-5209	94	17	ρ	ρ	NOUN
cana-5209	94	18	,	,	PUNCT
cana-5209	94	19	d)-pseudo	d)-pseudo	NOUN
cana-5209	94	20	-	-	NOUN
cana-5209	94	21	invex	invex	NOUN
cana-5209	94	22	on	on	ADP
cana-5209	94	23	x	x	PUNCT
cana-5209	94	24	iff	iff	PROPN
cana-5209	94	25	it	it	PRON
cana-5209	94	26	is	be	AUX
cana-5209	94	27	weak	weak	ADJ
cana-5209	94	28	strictly	strictly	ADV
cana-5209	94	29	(	(	PUNCT
cana-5209	94	30	c	c	X
cana-5209	94	31	,	,	PUNCT
cana-5209	94	32	α	α	PROPN
cana-5209	94	33	,	,	PUNCT
cana-5209	94	34	η	η	PROPN
cana-5209	94	35	,	,	PUNCT
cana-5209	94	36	ρ	ρ	NOUN
cana-5209	94	37	,	,	PUNCT
cana-5209	94	38	d)-pseudo	d)-pseudo	NOUN
cana-5209	94	39	-	-	NOUN
cana-5209	94	40	invex	invex	NOUN
cana-5209	94	41	at	at	ADP
cana-5209	94	42	each	each	DET
cana-5209	94	43	point	point	NOUN
cana-5209	94	44	in	in	ADP
cana-5209	94	45	x.	x.	NOUN
cana-5209	94	46	definition	definition	NOUN
cana-5209	94	47	3.4	3.4	NUM
cana-5209	94	48	.	.	PUNCT
cana-5209	95	1	the	the	DET
cana-5209	95	2	vector	vector	NOUN
cana-5209	95	3	-	-	PUNCT
cana-5209	95	4	valued	value	VERB
cana-5209	95	5	function	function	NOUN
cana-5209	95	6	f	f	NOUN
cana-5209	95	7	:	:	PUNCT
cana-5209	95	8	x	x	X
cana-5209	95	9	→	→	SYM
cana-5209	95	10	rp	rp	NOUN
cana-5209	95	11	is	be	AUX
cana-5209	95	12	said	say	VERB
cana-5209	95	13	to	to	PART
cana-5209	95	14	be	be	AUX
cana-5209	95	15	strong	strong	ADJ
cana-5209	95	16	(	(	PUNCT
cana-5209	95	17	c	c	X
cana-5209	95	18	,	,	PUNCT
cana-5209	95	19	α	α	PROPN
cana-5209	95	20	,	,	PUNCT
cana-5209	95	21	η	η	PROPN
cana-5209	95	22	,	,	PUNCT
cana-5209	95	23	ρ	ρ	PROPN
cana-5209	95	24	,	,	PUNCT
cana-5209	95	25	d)-pseudoinvex	d)-pseudoinvex	PUNCT
cana-5209	95	26	at	at	ADP
cana-5209	95	27	x0	x0	PROPN
cana-5209	95	28	∈	∈	PROPN
cana-5209	95	29	x	x	INTJ
cana-5209	95	30	if	if	SCONJ
cana-5209	95	31	,	,	PUNCT
cana-5209	95	32	for	for	ADP
cana-5209	95	33	each	each	DET
cana-5209	95	34	fi	fi	NOUN
cana-5209	95	35	:	:	PUNCT
cana-5209	95	36	x	x	X
cana-5209	95	37	→	→	SYM
cana-5209	95	38	r	r	NOUN
cana-5209	95	39	there	there	PRON
cana-5209	95	40	exists	exist	VERB
cana-5209	95	41	a	a	DET
cana-5209	95	42	vectorial	vectorial	ADJ
cana-5209	95	43	function	function	NOUN
cana-5209	95	44	ηi	ηi	NOUN
cana-5209	95	45	:	:	PUNCT
cana-5209	95	46	x	x	X
cana-5209	95	47	×x	×x	ADP
cana-5209	95	48	→	→	SYM
cana-5209	95	49	r	r	NOUN
cana-5209	95	50	such	such	ADJ
cana-5209	95	51	that	that	PRON
cana-5209	95	52	for	for	ADP
cana-5209	95	53	each	each	DET
cana-5209	95	54	x	x	SYM
cana-5209	95	55	∈	∈	PROPN
cana-5209	95	56	x	x	X
cana-5209	95	57	and	and	CCONJ
cana-5209	95	58	i	i	PRON
cana-5209	95	59	∈	∈	PROPN
cana-5209	95	60	p	p	X
cana-5209	95	61	,	,	PUNCT
cana-5209	95	62	the	the	DET
cana-5209	95	63	following	follow	VERB
cana-5209	95	64	condition	condition	NOUN
cana-5209	95	65	holds	hold	VERB
cana-5209	95	66	:	:	PUNCT
cana-5209	95	67	fi(x	fi(x	NUM
cana-5209	95	68	)	)	PUNCT
cana-5209	95	69	≤	≤	NUM
cana-5209	95	70	fi(x0	fi(x0	NOUN
cana-5209	95	71	)	)	PUNCT
cana-5209	95	72	⇒	⇒	PROPN
cana-5209	95	73	c(x	c(x	PROPN
cana-5209	95	74	,	,	PUNCT
cana-5209	95	75	x0)(∇fi(x0))ηi(x	x0)(∇fi(x0))ηi(x	NUM
cana-5209	95	76	,	,	PUNCT
cana-5209	95	77	x0	x0	PROPN
cana-5209	95	78	)	)	PUNCT
cana-5209	96	1	+	+	CCONJ
cana-5209	96	2	ρi	ρi	NOUN
cana-5209	96	3	di(x	di(x	NOUN
cana-5209	96	4	,	,	PUNCT
cana-5209	96	5	x0	x0	PROPN
cana-5209	96	6	)	)	PUNCT
cana-5209	96	7	αi(x	αi(x	NUM
cana-5209	96	8	,	,	PUNCT
cana-5209	96	9	x0	x0	PROPN
cana-5209	96	10	)	)	PUNCT
cana-5209	96	11	≤	≤	NUM
cana-5209	96	12	0	0	NUM
cana-5209	96	13	.	.	PUNCT
cana-5209	97	1	https://internationalpubls.com	https://internationalpubls.com	X
cana-5209	97	2	1142	1142	NUM
cana-5209	97	3	communications	communication	NOUN
cana-5209	97	4	on	on	ADP
cana-5209	97	5	applied	apply	VERB
cana-5209	97	6	nonlinear	nonlinear	ADJ
cana-5209	97	7	analysis	analysis	NOUN
cana-5209	97	8	issn	issn	NOUN
cana-5209	97	9	:	:	PUNCT
cana-5209	97	10	1074	1074	NUM
cana-5209	97	11	-	-	PUNCT
cana-5209	97	12	133x	133x	NUM
cana-5209	97	13	vol	vol	VERB
cana-5209	97	14	32	32	NUM
cana-5209	97	15	no	no	NOUN
cana-5209	97	16	.	.	PUNCT
cana-5209	98	1	10s	10	NOUN
cana-5209	98	2	(	(	PUNCT
cana-5209	98	3	2025	2025	NUM
cana-5209	98	4	)	)	PUNCT
cana-5209	98	5	the	the	DET
cana-5209	98	6	function	function	NOUN
cana-5209	98	7	f	f	PROPN
cana-5209	98	8	is	be	AUX
cana-5209	98	9	said	say	VERB
cana-5209	98	10	to	to	PART
cana-5209	98	11	be	be	AUX
cana-5209	98	12	strong	strong	ADJ
cana-5209	98	13	(	(	PUNCT
cana-5209	98	14	c	c	X
cana-5209	98	15	,	,	PUNCT
cana-5209	98	16	α	α	PROPN
cana-5209	98	17	,	,	PUNCT
cana-5209	98	18	η	η	PROPN
cana-5209	98	19	,	,	PUNCT
cana-5209	98	20	ρ	ρ	NOUN
cana-5209	98	21	,	,	PUNCT
cana-5209	98	22	d)-pseudo	d)-pseudo	NOUN
cana-5209	98	23	-	-	NOUN
cana-5209	98	24	invex	invex	NOUN
cana-5209	98	25	on	on	ADP
cana-5209	98	26	x	x	PUNCT
cana-5209	98	27	iff	iff	PROPN
cana-5209	98	28	it	it	PRON
cana-5209	98	29	is	be	AUX
cana-5209	98	30	strong	strong	ADJ
cana-5209	98	31	(	(	PUNCT
cana-5209	98	32	c	c	X
cana-5209	98	33	,	,	PUNCT
cana-5209	98	34	α	α	PROPN
cana-5209	98	35	,	,	PUNCT
cana-5209	98	36	η	η	PROPN
cana-5209	98	37	,	,	PUNCT
cana-5209	98	38	ρ	ρ	PROPN
cana-5209	98	39	,	,	PUNCT
cana-5209	98	40	d)pseudo	d)pseudo	NOUN
cana-5209	98	41	-	-	NOUN
cana-5209	98	42	invex	invex	NOUN
cana-5209	98	43	at	at	ADP
cana-5209	98	44	each	each	DET
cana-5209	98	45	point	point	NOUN
cana-5209	98	46	in	in	ADP
cana-5209	98	47	x.	x.	NOUN
cana-5209	98	48	definition	definition	NOUN
cana-5209	98	49	3.5	3.5	NUM
cana-5209	98	50	.	.	PUNCT
cana-5209	99	1	the	the	DET
cana-5209	99	2	vector	vector	NOUN
cana-5209	99	3	-	-	PUNCT
cana-5209	99	4	valued	value	VERB
cana-5209	99	5	function	function	NOUN
cana-5209	99	6	f	f	NOUN
cana-5209	99	7	:	:	PUNCT
cana-5209	99	8	x	x	X
cana-5209	99	9	→	→	SYM
cana-5209	99	10	rp	rp	NOUN
cana-5209	99	11	is	be	AUX
cana-5209	99	12	said	say	VERB
cana-5209	99	13	to	to	PART
cana-5209	99	14	be	be	AUX
cana-5209	99	15	(	(	PUNCT
cana-5209	99	16	c	c	X
cana-5209	99	17	,	,	PUNCT
cana-5209	99	18	α	α	PROPN
cana-5209	99	19	,	,	PUNCT
cana-5209	99	20	η	η	PROPN
cana-5209	99	21	,	,	PUNCT
cana-5209	99	22	ρ	ρ	PROPN
cana-5209	99	23	,	,	PUNCT
cana-5209	99	24	d)-quasi	d)-quasi	NOUN
cana-5209	99	25	-	-	PUNCT
cana-5209	99	26	invex	invex	NOUN
cana-5209	99	27	at	at	ADP
cana-5209	99	28	x0	x0	PROPN
cana-5209	99	29	∈	∈	PROPN
cana-5209	99	30	x	x	INTJ
cana-5209	99	31	if	if	SCONJ
cana-5209	99	32	,	,	PUNCT
cana-5209	99	33	for	for	ADP
cana-5209	99	34	each	each	DET
cana-5209	99	35	fi	fi	NOUN
cana-5209	99	36	:	:	PUNCT
cana-5209	99	37	x	x	X
cana-5209	99	38	→	→	SYM
cana-5209	99	39	r	r	NOUN
cana-5209	99	40	there	there	PRON
cana-5209	99	41	exists	exist	VERB
cana-5209	99	42	a	a	DET
cana-5209	99	43	vectorial	vectorial	ADJ
cana-5209	99	44	function	function	NOUN
cana-5209	99	45	ηi	ηi	NOUN
cana-5209	99	46	:	:	PUNCT
cana-5209	99	47	x	x	X
cana-5209	99	48	×x	×x	ADP
cana-5209	99	49	→	→	SYM
cana-5209	99	50	r	r	NOUN
cana-5209	99	51	such	such	ADJ
cana-5209	99	52	that	that	PRON
cana-5209	99	53	for	for	ADP
cana-5209	99	54	each	each	DET
cana-5209	99	55	x	x	SYM
cana-5209	99	56	∈	∈	PROPN
cana-5209	99	57	x	x	X
cana-5209	99	58	and	and	CCONJ
cana-5209	99	59	i	i	PRON
cana-5209	99	60	∈	∈	PROPN
cana-5209	99	61	p	p	X
cana-5209	99	62	,	,	PUNCT
cana-5209	99	63	the	the	DET
cana-5209	99	64	following	follow	VERB
cana-5209	99	65	condition	condition	NOUN
cana-5209	99	66	holds	hold	VERB
cana-5209	99	67	:	:	PUNCT
cana-5209	99	68	fi(x	fi(x	NUM
cana-5209	99	69	)	)	PUNCT
cana-5209	99	70	≦	≦	NUM
cana-5209	99	71	fi(x0	fi(x0	NOUN
cana-5209	99	72	)	)	PUNCT
cana-5209	99	73	⇒	⇒	PROPN
cana-5209	99	74	c(x	c(x	PROPN
cana-5209	99	75	,	,	PUNCT
cana-5209	99	76	x0)(∇fi(x0))ηi(x	x0)(∇fi(x0))ηi(x	NUM
cana-5209	99	77	,	,	PUNCT
cana-5209	99	78	x0	x0	PROPN
cana-5209	99	79	)	)	PUNCT
cana-5209	100	1	+	+	CCONJ
cana-5209	100	2	ρi	ρi	NOUN
cana-5209	100	3	di(x	di(x	NOUN
cana-5209	100	4	,	,	PUNCT
cana-5209	100	5	x0	x0	PROPN
cana-5209	100	6	)	)	PUNCT
cana-5209	100	7	αi(x	αi(x	NUM
cana-5209	100	8	,	,	PUNCT
cana-5209	100	9	x0	x0	PROPN
cana-5209	100	10	)	)	PUNCT
cana-5209	100	11	≦	≦	VERB
cana-5209	100	12	0	0	PUNCT
cana-5209	100	13	.	.	PUNCT
cana-5209	101	1	the	the	DET
cana-5209	101	2	function	function	NOUN
cana-5209	101	3	f	f	PROPN
cana-5209	101	4	is	be	AUX
cana-5209	101	5	said	say	VERB
cana-5209	101	6	to	to	PART
cana-5209	101	7	be	be	AUX
cana-5209	101	8	(	(	PUNCT
cana-5209	101	9	c	c	X
cana-5209	101	10	,	,	PUNCT
cana-5209	101	11	α	α	PROPN
cana-5209	101	12	,	,	PUNCT
cana-5209	101	13	η	η	PROPN
cana-5209	101	14	,	,	PUNCT
cana-5209	101	15	ρ	ρ	PROPN
cana-5209	101	16	,	,	PUNCT
cana-5209	101	17	d)-quasi	d)-quasi	NOUN
cana-5209	101	18	-	-	PUNCT
cana-5209	101	19	invex	invex	NOUN
cana-5209	101	20	on	on	ADP
cana-5209	101	21	x	x	PUNCT
cana-5209	101	22	iff	iff	PROPN
cana-5209	101	23	it	it	PRON
cana-5209	101	24	is	be	AUX
cana-5209	101	25	(	(	PUNCT
cana-5209	101	26	c	c	X
cana-5209	101	27	,	,	PUNCT
cana-5209	101	28	α	α	PROPN
cana-5209	101	29	,	,	PUNCT
cana-5209	101	30	η	η	PROPN
cana-5209	101	31	,	,	PUNCT
cana-5209	101	32	ρ	ρ	PROPN
cana-5209	101	33	,	,	PUNCT
cana-5209	101	34	d)-quasi	d)-quasi	NOUN
cana-5209	101	35	-	-	PUNCT
cana-5209	101	36	invex	invex	NOUN
cana-5209	101	37	at	at	ADP
cana-5209	101	38	each	each	DET
cana-5209	101	39	point	point	NOUN
cana-5209	101	40	in	in	ADP
cana-5209	101	41	x.	x.	NOUN
cana-5209	101	42	definition	definition	NOUN
cana-5209	101	43	3.6	3.6	NUM
cana-5209	101	44	.	.	PUNCT
cana-5209	102	1	the	the	DET
cana-5209	102	2	vector	vector	NOUN
cana-5209	102	3	-	-	PUNCT
cana-5209	102	4	valued	value	VERB
cana-5209	102	5	function	function	NOUN
cana-5209	102	6	f	f	NOUN
cana-5209	102	7	:	:	PUNCT
cana-5209	102	8	x	x	X
cana-5209	102	9	→	→	SYM
cana-5209	102	10	rp	rp	NOUN
cana-5209	102	11	is	be	AUX
cana-5209	102	12	said	say	VERB
cana-5209	102	13	to	to	PART
cana-5209	102	14	be	be	AUX
cana-5209	102	15	weak	weak	ADJ
cana-5209	102	16	(	(	PUNCT
cana-5209	102	17	c	c	X
cana-5209	102	18	,	,	PUNCT
cana-5209	102	19	α	α	PROPN
cana-5209	102	20	,	,	PUNCT
cana-5209	102	21	η	η	PROPN
cana-5209	102	22	,	,	PUNCT
cana-5209	102	23	ρ	ρ	PROPN
cana-5209	102	24	,	,	PUNCT
cana-5209	102	25	d)-quasi	d)-quasi	NOUN
cana-5209	102	26	-	-	PUNCT
cana-5209	102	27	invex	invex	NOUN
cana-5209	102	28	at	at	ADP
cana-5209	102	29	x0	x0	PROPN
cana-5209	102	30	∈	∈	PROPN
cana-5209	102	31	x	x	INTJ
cana-5209	102	32	if	if	SCONJ
cana-5209	102	33	,	,	PUNCT
cana-5209	102	34	for	for	ADP
cana-5209	102	35	each	each	DET
cana-5209	102	36	fi	fi	NOUN
cana-5209	102	37	:	:	PUNCT
cana-5209	102	38	x	x	X
cana-5209	102	39	→	→	SYM
cana-5209	102	40	r	r	NOUN
cana-5209	102	41	there	there	PRON
cana-5209	102	42	exists	exist	VERB
cana-5209	102	43	a	a	DET
cana-5209	102	44	vectorial	vectorial	ADJ
cana-5209	102	45	function	function	NOUN
cana-5209	102	46	ηi	ηi	NOUN
cana-5209	102	47	:	:	PUNCT
cana-5209	103	1	x	x	PUNCT
cana-5209	103	2	×	×	NOUN
cana-5209	103	3	x	x	INTJ
cana-5209	103	4	→	→	PUNCT
cana-5209	103	5	r	r	NOUN
cana-5209	103	6	such	such	ADJ
cana-5209	103	7	that	that	PRON
cana-5209	103	8	for	for	ADP
cana-5209	103	9	each	each	DET
cana-5209	103	10	x	x	SYM
cana-5209	103	11	∈	∈	PROPN
cana-5209	103	12	x	x	X
cana-5209	103	13	and	and	CCONJ
cana-5209	103	14	i	i	PRON
cana-5209	103	15	∈	∈	PROPN
cana-5209	103	16	p	p	X
cana-5209	103	17	,	,	PUNCT
cana-5209	103	18	the	the	DET
cana-5209	103	19	following	follow	VERB
cana-5209	103	20	condition	condition	NOUN
cana-5209	103	21	holds	hold	VERB
cana-5209	103	22	:	:	PUNCT
cana-5209	103	23	fi(x	fi(x	NUM
cana-5209	103	24	)	)	PUNCT
cana-5209	103	25	≤	≤	NUM
cana-5209	103	26	fi(x0	fi(x0	NOUN
cana-5209	103	27	)	)	PUNCT
cana-5209	103	28	⇒	⇒	PROPN
cana-5209	103	29	c(x	c(x	PROPN
cana-5209	103	30	,	,	PUNCT
cana-5209	103	31	x0)(∇fi(x0))ηi(x	x0)(∇fi(x0))ηi(x	NUM
cana-5209	103	32	,	,	PUNCT
cana-5209	103	33	x0	x0	PROPN
cana-5209	103	34	)	)	PUNCT
cana-5209	104	1	+	+	CCONJ
cana-5209	104	2	ρi	ρi	NOUN
cana-5209	104	3	di(x	di(x	NOUN
cana-5209	104	4	,	,	PUNCT
cana-5209	104	5	x0	x0	PROPN
cana-5209	104	6	)	)	PUNCT
cana-5209	104	7	αi(x	αi(x	NUM
cana-5209	104	8	,	,	PUNCT
cana-5209	104	9	x0	x0	PROPN
cana-5209	104	10	)	)	PUNCT
cana-5209	104	11	≦	≦	VERB
cana-5209	104	12	0	0	PUNCT
cana-5209	104	13	.	.	PUNCT
cana-5209	105	1	the	the	DET
cana-5209	105	2	function	function	NOUN
cana-5209	105	3	f	f	PROPN
cana-5209	105	4	is	be	AUX
cana-5209	105	5	said	say	VERB
cana-5209	105	6	to	to	PART
cana-5209	105	7	be	be	AUX
cana-5209	105	8	weak	weak	ADJ
cana-5209	105	9	(	(	PUNCT
cana-5209	105	10	c	c	X
cana-5209	105	11	,	,	PUNCT
cana-5209	105	12	α	α	PROPN
cana-5209	105	13	,	,	PUNCT
cana-5209	105	14	η	η	PROPN
cana-5209	105	15	,	,	PUNCT
cana-5209	105	16	ρ	ρ	PROPN
cana-5209	105	17	,	,	PUNCT
cana-5209	105	18	d)-quasi	d)-quasi	NOUN
cana-5209	105	19	-	-	PUNCT
cana-5209	105	20	invex	invex	NOUN
cana-5209	105	21	on	on	ADP
cana-5209	105	22	x	x	SYM
cana-5209	105	23	if	if	SCONJ
cana-5209	105	24	and	and	CCONJ
cana-5209	105	25	only	only	ADV
cana-5209	105	26	if	if	SCONJ
cana-5209	105	27	it	it	PRON
cana-5209	105	28	is	be	AUX
cana-5209	105	29	weak	weak	ADJ
cana-5209	105	30	(	(	PUNCT
cana-5209	105	31	c	c	X
cana-5209	105	32	,	,	PUNCT
cana-5209	105	33	α	α	PROPN
cana-5209	105	34	,	,	PUNCT
cana-5209	105	35	η	η	PROPN
cana-5209	105	36	,	,	PUNCT
cana-5209	105	37	ρ	ρ	PROPN
cana-5209	105	38	,	,	PUNCT
cana-5209	105	39	d)quasi	d)quasi	NOUN
cana-5209	105	40	-	-	PUNCT
cana-5209	105	41	invex	invex	NOUN
cana-5209	105	42	at	at	ADP
cana-5209	105	43	each	each	DET
cana-5209	105	44	point	point	NOUN
cana-5209	105	45	in	in	ADP
cana-5209	105	46	x.	x.	NOUN
cana-5209	105	47	4	4	NUM
cana-5209	105	48	optimality	optimality	NOUN
cana-5209	105	49	for	for	ADP
cana-5209	105	50	the	the	DET
cana-5209	105	51	nonlinear	nonlinear	ADJ
cana-5209	105	52	multiobjective	multiobjective	ADJ
cana-5209	105	53	semiinfinite	semiinfinite	ADJ
cana-5209	105	54	programming	programming	NOUN
cana-5209	105	55	problem	problem	NOUN
cana-5209	105	56	(	(	PUNCT
cana-5209	105	57	sp	sp	NOUN
cana-5209	105	58	)	)	PUNCT
cana-5209	105	59	,	,	PUNCT
cana-5209	105	60	we	we	PRON
cana-5209	105	61	state	state	VERB
cana-5209	105	62	the	the	DET
cana-5209	105	63	following	follow	VERB
cana-5209	105	64	optimality	optimality	NOUN
cana-5209	105	65	conditions	condition	NOUN
cana-5209	105	66	.	.	PUNCT
cana-5209	106	1	theorem	theorem	VERB
cana-5209	106	2	4.1	4.1	NUM
cana-5209	106	3	.	.	PUNCT
cana-5209	107	1	if	if	SCONJ
cana-5209	107	2	there	there	PRON
cana-5209	107	3	exist	exist	VERB
cana-5209	107	4	a	a	DET
cana-5209	107	5	feasible	feasible	ADJ
cana-5209	107	6	solution	solution	NOUN
cana-5209	107	7	x0	x0	PROPN
cana-5209	107	8	for	for	ADP
cana-5209	107	9	(	(	PUNCT
cana-5209	107	10	sp	sp	NOUN
cana-5209	107	11	)	)	PUNCT
cana-5209	107	12	and	and	CCONJ
cana-5209	107	13	,	,	PUNCT
cana-5209	107	14	u	u	PROPN
cana-5209	107	15	∈	∈	PROPN
cana-5209	107	16	rp	rp	NOUN
cana-5209	107	17	and	and	CCONJ
cana-5209	107	18	v	v	NOUN
cana-5209	107	19	=	=	PUNCT
cana-5209	107	20	(	(	PUNCT
cana-5209	107	21	vj)j∈j	vj)j∈j	NUM
cana-5209	107	22	are	be	AUX
cana-5209	107	23	vectors	vector	NOUN
cana-5209	107	24	which	which	PRON
cana-5209	107	25	satisfies	satisfy	VERB
cana-5209	107	26	ut∇f(x0	ut∇f(x0	ADJ
cana-5209	107	27	)	)	PUNCT
cana-5209	108	1	+	+	CCONJ
cana-5209	108	2	vti	vti	PROPN
cana-5209	108	3	∇gi(x0	∇gi(x0	PROPN
cana-5209	108	4	)	)	PUNCT
cana-5209	108	5	=	=	SYM
cana-5209	108	6	0	0	NUM
cana-5209	108	7	,	,	PUNCT
cana-5209	108	8	vt	vt	NOUN
cana-5209	108	9	g(x0	g(x0	NOUN
cana-5209	108	10	)	)	PUNCT
cana-5209	108	11	=	=	SYM
cana-5209	108	12	0	0	NUM
cana-5209	108	13	,	,	PUNCT
cana-5209	108	14	(	(	PUNCT
cana-5209	108	15	4.1	4.1	NUM
cana-5209	108	16	)	)	PUNCT
cana-5209	108	17	u	u	NOUN
cana-5209	108	18	≥	≥	NOUN
cana-5209	108	19	0	0	NUM
cana-5209	108	20	,	,	PUNCT
cana-5209	108	21	v	v	PRON
cana-5209	108	22	≥	≥	NOUN
cana-5209	108	23	0	0	NUM
cana-5209	108	24	and	and	CCONJ
cana-5209	108	25	vj	vj	INTJ
cana-5209	108	26	̸=	̸=	PROPN
cana-5209	108	27	0	0	NUM
cana-5209	108	28	for	for	ADP
cana-5209	108	29	finite	finite	NOUN
cana-5209	108	30	many	many	ADJ
cana-5209	108	31	j	j	PROPN
cana-5209	108	32	∈	∈	PROPN
cana-5209	108	33	i.	i.	NOUN
cana-5209	108	34	also	also	ADV
cana-5209	108	35	,	,	PUNCT
cana-5209	108	36	if	if	SCONJ
cana-5209	108	37	f	f	PROPN
cana-5209	108	38	is	be	AUX
cana-5209	108	39	strong	strong	ADJ
cana-5209	108	40	(	(	PUNCT
cana-5209	108	41	c	c	X
cana-5209	108	42	,	,	PUNCT
cana-5209	108	43	α1	α1	PROPN
cana-5209	108	44	,	,	PUNCT
cana-5209	108	45	η1	η1	NOUN
cana-5209	108	46	,	,	PUNCT
cana-5209	108	47	ρ1	ρ1	NOUN
cana-5209	108	48	,	,	PUNCT
cana-5209	108	49	d1)-pseudoinvex	d1)-pseudoinvex	NOUN
cana-5209	108	50	function	function	NOUN
cana-5209	108	51	at	at	ADP
cana-5209	108	52	x0	x0	PROPN
cana-5209	108	53	and	and	CCONJ
cana-5209	108	54	gi	gi	PROPN
cana-5209	108	55	is	be	AUX
cana-5209	108	56	(	(	PUNCT
cana-5209	108	57	c	c	NOUN
cana-5209	108	58	,	,	PUNCT
cana-5209	108	59	α2	α2	ADJ
cana-5209	108	60	,	,	PUNCT
cana-5209	108	61	η2	η2	NOUN
cana-5209	108	62	,	,	PUNCT
cana-5209	108	63	ρ2	ρ2	NOUN
cana-5209	108	64	,	,	PUNCT
cana-5209	108	65	d2)-quasiinvex	d2)-quasiinvex	NOUN
cana-5209	108	66	function	function	NOUN
cana-5209	108	67	at	at	ADP
cana-5209	108	68	x0	x0	PROPN
cana-5209	108	69	w.r.t	w.r.t	NOUN
cana-5209	108	70	.	.	PUNCT
cana-5209	109	1	the	the	DET
cana-5209	109	2	same	same	ADJ
cana-5209	109	3	η(x	η(x	PROPN
cana-5209	109	4	,	,	PUNCT
cana-5209	109	5	x0	x0	PROPN
cana-5209	109	6	)	)	PUNCT
cana-5209	109	7	(	(	PUNCT
cana-5209	109	8	i.e.	i.e.	X
cana-5209	109	9	,	,	PUNCT
cana-5209	109	10	η	η	PROPN
cana-5209	109	11	1	1	NUM
cana-5209	109	12	=	=	SYM
cana-5209	109	13	η2	η2	X
cana-5209	109	14	=	=	SYM
cana-5209	109	15	η	η	NOUN
cana-5209	109	16	)	)	PUNCT
cana-5209	109	17	with	with	ADP
cana-5209	109	18	p∑	p∑	PROPN
cana-5209	109	19	i=1	i=1	PRON
cana-5209	109	20	uiρ	uiρ	ADJ
cana-5209	109	21	1	1	NUM
cana-5209	109	22	i	i	PRON
cana-5209	109	23	d1i	d1i	VERB
cana-5209	109	24	(	(	PUNCT
cana-5209	109	25	x	x	X
cana-5209	109	26	,	,	PUNCT
cana-5209	109	27	x0	x0	PROPN
cana-5209	109	28	)	)	PUNCT
cana-5209	109	29	α1	α1	PROPN
cana-5209	109	30	i	i	PRON
cana-5209	109	31	(	(	PUNCT
cana-5209	109	32	x	x	NOUN
cana-5209	109	33	,	,	PUNCT
cana-5209	109	34	x0	x0	PROPN
cana-5209	109	35	)	)	PUNCT
cana-5209	110	1	+	+	CCONJ
cana-5209	110	2	∑	∑	AUX
cana-5209	110	3	j∈i	j∈i	PROPN
cana-5209	110	4	vjρ	vjρ	VERB
cana-5209	110	5	2	2	NUM
cana-5209	110	6	j	j	PROPN
cana-5209	110	7	d2j	d2j	PROPN
cana-5209	110	8	(	(	PUNCT
cana-5209	110	9	x	x	NOUN
cana-5209	110	10	,	,	PUNCT
cana-5209	110	11	x0	x0	PROPN
cana-5209	110	12	)	)	PUNCT
cana-5209	110	13	α2	α2	PROPN
cana-5209	110	14	j	j	PROPN
cana-5209	110	15	(	(	PUNCT
cana-5209	110	16	x	x	PROPN
cana-5209	110	17	,	,	PUNCT
cana-5209	110	18	x0	x0	PROPN
cana-5209	110	19	)	)	PUNCT
cana-5209	110	20	≧	≧	X
cana-5209	110	21	0	0	X
cana-5209	110	22	.	.	PUNCT
cana-5209	111	1	(	(	PUNCT
cana-5209	111	2	4.2	4.2	NUM
cana-5209	111	3	)	)	PUNCT
cana-5209	111	4	then	then	ADV
cana-5209	111	5	,	,	PUNCT
cana-5209	111	6	x0	x0	PROPN
cana-5209	111	7	is	be	AUX
cana-5209	111	8	an	an	DET
cana-5209	111	9	efficient	efficient	ADJ
cana-5209	111	10	solution	solution	NOUN
cana-5209	111	11	of	of	ADP
cana-5209	111	12	(	(	PUNCT
cana-5209	111	13	sp	sp	NOUN
cana-5209	111	14	)	)	PUNCT
cana-5209	111	15	proof	proof	NOUN
cana-5209	111	16	.	.	PUNCT
cana-5209	112	1	assume	assume	VERB
cana-5209	112	2	that	that	SCONJ
cana-5209	112	3	x0	x0	PROPN
cana-5209	112	4	is	be	AUX
cana-5209	112	5	not	not	PART
cana-5209	112	6	an	an	DET
cana-5209	112	7	efficient	efficient	ADJ
cana-5209	112	8	solution	solution	NOUN
cana-5209	112	9	of	of	ADP
cana-5209	112	10	(	(	PUNCT
cana-5209	112	11	sp	sp	NOUN
cana-5209	112	12	)	)	PUNCT
cana-5209	112	13	,	,	PUNCT
cana-5209	112	14	then	then	ADV
cana-5209	112	15	for	for	ADP
cana-5209	112	16	a	a	DET
cana-5209	112	17	feasible	feasible	ADJ
cana-5209	112	18	solution	solution	NOUN
cana-5209	112	19	x	x	X
cana-5209	112	20	∈	∈	PROPN
cana-5209	112	21	s	s	X
cana-5209	112	22	,	,	PUNCT
cana-5209	112	23	we	we	PRON
cana-5209	112	24	have	have	AUX
cana-5209	112	25	fi(x	fi(x	NUM
cana-5209	112	26	)	)	PUNCT
cana-5209	112	27	≤	≤	NUM
cana-5209	112	28	fi(x0	fi(x0	NOUN
cana-5209	112	29	)	)	PUNCT
cana-5209	112	30	.	.	PUNCT
cana-5209	113	1	since	since	SCONJ
cana-5209	113	2	gi(x0	gi(x0	NOUN
cana-5209	113	3	)	)	PUNCT
cana-5209	113	4	=	=	SYM
cana-5209	113	5	0	0	NUM
cana-5209	113	6	,	,	PUNCT
cana-5209	113	7	hence	hence	ADV
cana-5209	113	8	gi(x	gi(x	PROPN
cana-5209	113	9	)	)	PUNCT
cana-5209	113	10	≦	≦	NUM
cana-5209	113	11	gi(x0	gi(x0	NOUN
cana-5209	113	12	)	)	PUNCT
cana-5209	113	13	.	.	PUNCT
cana-5209	114	1	since	since	SCONJ
cana-5209	114	2	f	f	PROPN
cana-5209	114	3	is	be	AUX
cana-5209	114	4	strong	strong	ADJ
cana-5209	114	5	(	(	PUNCT
cana-5209	114	6	c	c	X
cana-5209	114	7	,	,	PUNCT
cana-5209	114	8	α1	α1	PROPN
cana-5209	114	9	,	,	PUNCT
cana-5209	114	10	η1	η1	NOUN
cana-5209	114	11	,	,	PUNCT
cana-5209	114	12	ρ1	ρ1	NOUN
cana-5209	114	13	,	,	PUNCT
cana-5209	114	14	d1)-pseudo	d1)-pseudo	NOUN
cana-5209	114	15	-	-	NOUN
cana-5209	114	16	invex	invex	NOUN
cana-5209	114	17	at	at	ADP
cana-5209	114	18	x0	x0	PROPN
cana-5209	114	19	and	and	CCONJ
cana-5209	114	20	gi	gi	PROPN
cana-5209	114	21	is	be	AUX
cana-5209	114	22	(	(	PUNCT
cana-5209	114	23	c	c	NOUN
cana-5209	114	24	,	,	PUNCT
cana-5209	114	25	α2	α2	ADJ
cana-5209	114	26	,	,	PUNCT
cana-5209	114	27	η2	η2	NOUN
cana-5209	114	28	,	,	PUNCT
cana-5209	114	29	ρ2	ρ2	NOUN
cana-5209	114	30	,	,	PUNCT
cana-5209	114	31	d2)-quasi	d2)-quasi	NOUN
cana-5209	114	32	-	-	NOUN
cana-5209	114	33	invex	invex	NOUN
cana-5209	114	34	at	at	ADP
cana-5209	114	35	x0	x0	PROPN
cana-5209	114	36	,	,	PUNCT
cana-5209	114	37	therefore	therefore	ADV
cana-5209	114	38	we	we	PRON
cana-5209	114	39	have	have	VERB
cana-5209	114	40	c(x	c(x	NOUN
cana-5209	114	41	,	,	PUNCT
cana-5209	114	42	x0)(∇fi(x0))η(x	x0)(∇fi(x0))η(x	PROPN
cana-5209	114	43	,	,	PUNCT
cana-5209	114	44	x0	x0	PROPN
cana-5209	114	45	)	)	PUNCT
cana-5209	115	1	+	+	NUM
cana-5209	115	2	ρ1i	ρ1i	PUNCT
cana-5209	115	3	d1i	d1i	NOUN
cana-5209	115	4	(	(	PUNCT
cana-5209	115	5	x	x	X
cana-5209	115	6	,	,	PUNCT
cana-5209	115	7	x0	x0	PROPN
cana-5209	115	8	)	)	PUNCT
cana-5209	115	9	α1	α1	PROPN
cana-5209	115	10	i	i	PRON
cana-5209	115	11	(	(	PUNCT
cana-5209	115	12	x	x	NOUN
cana-5209	115	13	,	,	PUNCT
cana-5209	115	14	x0	x0	PROPN
cana-5209	115	15	)	)	PUNCT
cana-5209	115	16	≤	≤	NOUN
cana-5209	115	17	0	0	NUM
cana-5209	115	18	,	,	PUNCT
cana-5209	115	19	and	and	CCONJ
cana-5209	115	20	c(x	c(x	NOUN
cana-5209	115	21	,	,	PUNCT
cana-5209	115	22	x0)(∇gi(x0))η(x	x0)(∇gi(x0))η(x	PROPN
cana-5209	115	23	,	,	PUNCT
cana-5209	115	24	x0	x0	PROPN
cana-5209	115	25	)	)	PUNCT
cana-5209	116	1	+	+	CCONJ
cana-5209	116	2	ρ2i	ρ2i	ADJ
cana-5209	116	3	d2i(x	d2i(x	PROPN
cana-5209	116	4	,	,	PUNCT
cana-5209	116	5	x0	x0	PROPN
cana-5209	116	6	)	)	PUNCT
cana-5209	116	7	α2	α2	ADJ
cana-5209	116	8	i(x	i(x	PROPN
cana-5209	116	9	,	,	PUNCT
cana-5209	116	10	x0	x0	PROPN
cana-5209	116	11	)	)	PUNCT
cana-5209	116	12	≦	≦	VERB
cana-5209	116	13	0	0	PUNCT
cana-5209	116	14	.	.	PUNCT
cana-5209	117	1	since	since	SCONJ
cana-5209	117	2	c	c	PROPN
cana-5209	117	3	is	be	AUX
cana-5209	117	4	convex	convex	ADJ
cana-5209	117	5	,	,	PUNCT
cana-5209	117	6	thus	thus	ADV
cana-5209	117	7	from	from	ADP
cana-5209	117	8	the	the	DET
cana-5209	117	9	above	above	ADJ
cana-5209	117	10	inequalities	inequality	NOUN
cana-5209	117	11	,	,	PUNCT
cana-5209	117	12	we	we	PRON
cana-5209	117	13	get	get	VERB
cana-5209	117	14	c(x	c(x	NOUN
cana-5209	117	15	,	,	PUNCT
cana-5209	117	16	x0	x0	PROPN
cana-5209	117	17	)	)	PUNCT
cana-5209	118	1			PROPN
cana-5209	118	2	p∑	p∑	NOUN
cana-5209	118	3	i=1	i=1	PROPN
cana-5209	118	4	1	1	NUM
cana-5209	118	5	τ	τ	NOUN
cana-5209	118	6	ui∇fi(x0	ui∇fi(x0	NOUN
cana-5209	118	7	)	)	PUNCT
cana-5209	119	1	+	+	CCONJ
cana-5209	119	2	∑	∑	PUNCT
cana-5209	119	3	j∈i	j∈i	PROPN
cana-5209	119	4	1	1	NUM
cana-5209	119	5	τ	τ	PROPN
cana-5209	119	6	vj∇gj(x0	vj∇gj(x0	NOUN
cana-5209	119	7	)	)	PUNCT
cana-5209	119	8			PROPN
cana-5209	119	9	η(x	η(x	NOUN
cana-5209	119	10	,	,	PUNCT
cana-5209	119	11	x0	x0	PROPN
cana-5209	119	12	)	)	PUNCT
cana-5209	119	13	https://internationalpubls.com	https://internationalpubls.com	X
cana-5209	119	14	1143	1143	NUM
cana-5209	119	15	communications	communication	NOUN
cana-5209	119	16	on	on	ADP
cana-5209	119	17	applied	apply	VERB
cana-5209	119	18	nonlinear	nonlinear	ADJ
cana-5209	119	19	analysis	analysis	NOUN
cana-5209	119	20	issn	issn	NOUN
cana-5209	119	21	:	:	PUNCT
cana-5209	119	22	1074	1074	NUM
cana-5209	119	23	-	-	PUNCT
cana-5209	119	24	133x	133x	NUM
cana-5209	119	25	vol	vol	VERB
cana-5209	119	26	32	32	NUM
cana-5209	119	27	no	no	NOUN
cana-5209	119	28	.	.	PUNCT
cana-5209	120	1	10s	10	NOUN
cana-5209	120	2	(	(	PUNCT
cana-5209	120	3	2025	2025	NUM
cana-5209	120	4	)	)	PUNCT
cana-5209	121	1	+	+	CCONJ
cana-5209	121	2	p∑	p∑	X
cana-5209	121	3	i=1	i=1	PROPN
cana-5209	121	4	1	1	NUM
cana-5209	121	5	τ	τ	NOUN
cana-5209	121	6	uiρ	uiρ	NOUN
cana-5209	121	7	1	1	NUM
cana-5209	121	8	i	i	PRON
cana-5209	121	9	d1i	d1i	VERB
cana-5209	121	10	(	(	PUNCT
cana-5209	121	11	x	x	X
cana-5209	121	12	,	,	PUNCT
cana-5209	121	13	x0	x0	PROPN
cana-5209	121	14	)	)	PUNCT
cana-5209	121	15	α1	α1	PROPN
cana-5209	121	16	i	i	PRON
cana-5209	121	17	(	(	PUNCT
cana-5209	121	18	x	x	NOUN
cana-5209	121	19	,	,	PUNCT
cana-5209	121	20	x0	x0	PROPN
cana-5209	121	21	)	)	PUNCT
cana-5209	122	1	+	+	CCONJ
cana-5209	122	2	∑	∑	PUNCT
cana-5209	122	3	j∈i	j∈i	PROPN
cana-5209	122	4	1	1	NUM
cana-5209	122	5	τ	τ	PROPN
cana-5209	122	6	vjρ	vjρ	PROPN
cana-5209	122	7	2	2	NUM
cana-5209	122	8	j	j	PROPN
cana-5209	122	9	d2j	d2j	PROPN
cana-5209	122	10	(	(	PUNCT
cana-5209	122	11	x	x	NOUN
cana-5209	122	12	,	,	PUNCT
cana-5209	122	13	x0	x0	PROPN
cana-5209	122	14	)	)	PUNCT
cana-5209	122	15	α2	α2	PROPN
cana-5209	122	16	j	j	PROPN
cana-5209	122	17	(	(	PUNCT
cana-5209	122	18	x	x	PROPN
cana-5209	122	19	,	,	PUNCT
cana-5209	122	20	x0	x0	PROPN
cana-5209	122	21	)	)	PUNCT
cana-5209	122	22	<	<	X
cana-5209	122	23	0	0	NUM
cana-5209	122	24	,	,	PUNCT
cana-5209	122	25	where	where	SCONJ
cana-5209	122	26	τ	τ	X
cana-5209	122	27	=	=	PUNCT
cana-5209	122	28	p∑	p∑	X
cana-5209	123	1	i=1	i=1	X
cana-5209	123	2	ui	ui	PROPN
cana-5209	124	1	+	+	CCONJ
cana-5209	124	2	∑	∑	PROPN
cana-5209	124	3	j∈i	j∈i	PROPN
cana-5209	124	4	vj	vj	INTJ
cana-5209	124	5	.	.	PUNCT
cana-5209	125	1	since	since	SCONJ
cana-5209	125	2	c(x	c(x	NOUN
cana-5209	125	3	,	,	PUNCT
cana-5209	125	4	x0)(0	x0)(0	PRON
cana-5209	125	5	)	)	PUNCT
cana-5209	125	6	=	=	SYM
cana-5209	125	7	0	0	NUM
cana-5209	125	8	,	,	PUNCT
cana-5209	125	9	thus	thus	ADV
cana-5209	125	10	from	from	ADP
cana-5209	125	11	equation	equation	NOUN
cana-5209	125	12	(	(	PUNCT
cana-5209	125	13	4.1	4.1	NUM
cana-5209	125	14	)	)	PUNCT
cana-5209	125	15	we	we	PRON
cana-5209	125	16	have	have	VERB
cana-5209	125	17	p∑	p∑	VERB
cana-5209	126	1	i=1	i=1	PRON
cana-5209	126	2	uiρ	uiρ	ADJ
cana-5209	127	1	1	1	NUM
cana-5209	127	2	i	i	PRON
cana-5209	127	3	d1i	d1i	VERB
cana-5209	127	4	(	(	PUNCT
cana-5209	127	5	x	x	X
cana-5209	127	6	,	,	PUNCT
cana-5209	127	7	x0	x0	PROPN
cana-5209	127	8	)	)	PUNCT
cana-5209	127	9	α1	α1	PROPN
cana-5209	128	1	i	i	PRON
cana-5209	128	2	(	(	PUNCT
cana-5209	128	3	x	x	NOUN
cana-5209	128	4	,	,	PUNCT
cana-5209	128	5	x0	x0	PROPN
cana-5209	128	6	)	)	PUNCT
cana-5209	129	1	+	+	CCONJ
cana-5209	129	2	∑	∑	AUX
cana-5209	129	3	j∈i	j∈i	PROPN
cana-5209	129	4	vjρ	vjρ	VERB
cana-5209	129	5	2	2	NUM
cana-5209	129	6	j	j	PROPN
cana-5209	129	7	d2j	d2j	PROPN
cana-5209	129	8	(	(	PUNCT
cana-5209	129	9	x	x	NOUN
cana-5209	129	10	,	,	PUNCT
cana-5209	129	11	x0	x0	PROPN
cana-5209	129	12	)	)	PUNCT
cana-5209	129	13	α2	α2	PROPN
cana-5209	129	14	j	j	PROPN
cana-5209	129	15	(	(	PUNCT
cana-5209	129	16	x	x	PROPN
cana-5209	129	17	,	,	PUNCT
cana-5209	129	18	x0	x0	PROPN
cana-5209	129	19	)	)	PUNCT
cana-5209	129	20	<	<	X
cana-5209	129	21	0	0	NUM
cana-5209	129	22	,	,	PUNCT
cana-5209	129	23	which	which	PRON
cana-5209	129	24	go	go	VERB
cana-5209	129	25	against	against	ADP
cana-5209	129	26	our	our	PRON
cana-5209	129	27	assumption	assumption	NOUN
cana-5209	129	28	(	(	PUNCT
cana-5209	129	29	4.2	4.2	NUM
cana-5209	129	30	)	)	PUNCT
cana-5209	129	31	.	.	PUNCT
cana-5209	130	1	thus	thus	ADV
cana-5209	130	2	x0	x0	PROPN
cana-5209	130	3	is	be	AUX
cana-5209	130	4	an	an	DET
cana-5209	130	5	efficient	efficient	ADJ
cana-5209	130	6	solution	solution	NOUN
cana-5209	130	7	of	of	ADP
cana-5209	130	8	(	(	PUNCT
cana-5209	130	9	sp	sp	NOUN
cana-5209	130	10	)	)	PUNCT
cana-5209	130	11	.	.	PUNCT
cana-5209	131	1	example	example	NOUN
cana-5209	131	2	:	:	PUNCT
cana-5209	131	3	consider	consider	VERB
cana-5209	131	4	the	the	DET
cana-5209	131	5	semiinfinite	semiinfinite	ADJ
cana-5209	131	6	problem	problem	NOUN
cana-5209	131	7	as	as	SCONJ
cana-5209	131	8	follows	follow	VERB
cana-5209	131	9	:	:	PUNCT
cana-5209	131	10	(	(	PUNCT
cana-5209	131	11	sp	sp	ADP
cana-5209	131	12	1	1	NUM
cana-5209	131	13	)	)	PUNCT
cana-5209	131	14	minimum	minimum	ADJ
cana-5209	131	15	f(x	f(x	PROPN
cana-5209	131	16	)	)	PUNCT
cana-5209	132	1	s.t	s.t	PROPN
cana-5209	132	2	.	.	PROPN
cana-5209	132	3	gj(x	gj(x	PUNCT
cana-5209	132	4	)	)	PUNCT
cana-5209	132	5	≦	≦	NUM
cana-5209	132	6	0	0	NUM
cana-5209	132	7	,	,	PUNCT
cana-5209	132	8	j	j	PROPN
cana-5209	132	9	∈	∈	PROPN
cana-5209	132	10	j	j	PROPN
cana-5209	132	11	,	,	PUNCT
cana-5209	132	12	x	x	PUNCT
cana-5209	132	13	∈	∈	PROPN
cana-5209	132	14	r	r	NOUN
cana-5209	132	15	,	,	PUNCT
cana-5209	132	16	where	where	SCONJ
cana-5209	132	17	the	the	DET
cana-5209	132	18	functions	function	NOUN
cana-5209	132	19	f	f	X
cana-5209	132	20	:	:	PUNCT
cana-5209	132	21	x	x	X
cana-5209	132	22	→	→	SYM
cana-5209	132	23	r2	r2	PROPN
cana-5209	132	24	and	and	CCONJ
cana-5209	132	25	gj	gj	NOUN
cana-5209	132	26	:	:	PUNCT
cana-5209	132	27	x	x	X
cana-5209	132	28	→	→	SYM
cana-5209	132	29	r	r	NOUN
cana-5209	132	30	,	,	PUNCT
cana-5209	132	31	(	(	PUNCT
cana-5209	132	32	x	x	X
cana-5209	132	33	=	=	SYM
cana-5209	132	34	r	r	NOUN
cana-5209	132	35	)	)	PUNCT
cana-5209	132	36	are	be	AUX
cana-5209	132	37	defined	define	VERB
cana-5209	132	38	as	as	ADP
cana-5209	132	39	:	:	PUNCT
cana-5209	132	40	f	f	PROPN
cana-5209	132	41	(	(	PUNCT
cana-5209	132	42	x	x	X
cana-5209	132	43	)	)	PUNCT
cana-5209	132	44	=	=	SYM
cana-5209	132	45	(	(	PUNCT
cana-5209	132	46	f1(x	f1(x	NOUN
cana-5209	132	47	)	)	PUNCT
cana-5209	132	48	,	,	PUNCT
cana-5209	132	49	f2(x	f2(x	NOUN
cana-5209	132	50	)	)	PUNCT
cana-5209	132	51	)	)	PUNCT
cana-5209	133	1	=	=	SYM
cana-5209	134	1	(	(	PUNCT
cana-5209	134	2	x2	x2	INTJ
cana-5209	134	3	−	−	PROPN
cana-5209	134	4	2x	2x	NUM
cana-5209	134	5	,	,	PUNCT
cana-5209	134	6	x3	x3	ADJ
cana-5209	134	7	−	−	PROPN
cana-5209	134	8	x2	x2	PROPN
cana-5209	134	9	)	)	PUNCT
cana-5209	134	10	and	and	CCONJ
cana-5209	134	11	g1(x	g1(x	NOUN
cana-5209	134	12	)	)	PUNCT
cana-5209	134	13	=	=	SYM
cana-5209	134	14	x2(x−	x2(x−	PROPN
cana-5209	134	15	2	2	NUM
cana-5209	134	16	)	)	PUNCT
cana-5209	134	17	,	,	PUNCT
cana-5209	134	18	g2(x	g2(x	X
cana-5209	134	19	)	)	PUNCT
cana-5209	134	20	=	=	PUNCT
cana-5209	135	1	x3	x3	ADJ
cana-5209	135	2	+	+	CCONJ
cana-5209	135	3	x	x	X
cana-5209	135	4	,	,	PUNCT
cana-5209	135	5	gk(x	gk(x	X
cana-5209	135	6	)	)	PUNCT
cana-5209	135	7	=	=	PUNCT
cana-5209	136	1	x+	x+	SYM
cana-5209	136	2	1	1	NUM
cana-5209	136	3	k	k	NOUN
cana-5209	136	4	,	,	PUNCT
cana-5209	136	5	k	k	PROPN
cana-5209	136	6	=	=	SYM
cana-5209	136	7	3	3	NUM
cana-5209	136	8	,	,	PUNCT
cana-5209	136	9	4	4	NUM
cana-5209	136	10	,	,	PUNCT
cana-5209	136	11	...	...	PUNCT
cana-5209	136	12	also	also	ADV
cana-5209	136	13	the	the	DET
cana-5209	136	14	function	function	NOUN
cana-5209	136	15	c	c	NOUN
cana-5209	136	16	:	:	PUNCT
cana-5209	137	1	r	r	NOUN
cana-5209	137	2	×	×	NOUN
cana-5209	137	3	r	r	NOUN
cana-5209	137	4	×	×	NOUN
cana-5209	137	5	r	r	NOUN
cana-5209	137	6	→	→	SYM
cana-5209	137	7	r	r	NOUN
cana-5209	137	8	is	be	AUX
cana-5209	137	9	defined	define	VERB
cana-5209	137	10	by	by	ADP
cana-5209	137	11	c(x	c(x	PROPN
cana-5209	137	12	,	,	PUNCT
cana-5209	137	13	x0)(a	x0)(a	NUM
cana-5209	137	14	)	)	PUNCT
cana-5209	138	1	=	=	SYM
cana-5209	139	1	−a2(x	−a2(x	PROPN
cana-5209	139	2	+	+	CCONJ
cana-5209	139	3	x0	x0	PROPN
cana-5209	139	4	)	)	PUNCT
cana-5209	139	5	,	,	PUNCT
cana-5209	139	6	the	the	DET
cana-5209	139	7	function	function	NOUN
cana-5209	139	8	η	η	PROPN
cana-5209	139	9	:	:	PUNCT
cana-5209	139	10	x	x	PROPN
cana-5209	139	11	×x	×x	ADP
cana-5209	139	12	→	→	SYM
cana-5209	139	13	r	r	NOUN
cana-5209	139	14	is	be	AUX
cana-5209	139	15	defined	define	VERB
cana-5209	139	16	by	by	ADP
cana-5209	139	17	η(x	η(x	NOUN
cana-5209	139	18	,	,	PUNCT
cana-5209	139	19	x0	x0	PROPN
cana-5209	139	20	)	)	PUNCT
cana-5209	140	1	=	=	SYM
cana-5209	140	2	(	(	PUNCT
cana-5209	140	3	2x	2x	NUM
cana-5209	140	4	−	−	NOUN
cana-5209	140	5	x0	x0	PROPN
cana-5209	140	6	)	)	PUNCT
cana-5209	140	7	and	and	CCONJ
cana-5209	140	8	the	the	DET
cana-5209	140	9	set	set	NOUN
cana-5209	140	10	s	s	X
cana-5209	140	11	of	of	ADP
cana-5209	140	12	feasible	feasible	ADJ
cana-5209	140	13	solutions	solution	NOUN
cana-5209	140	14	for	for	ADP
cana-5209	140	15	(	(	PUNCT
cana-5209	140	16	sp	sp	ADP
cana-5209	140	17	1	1	NUM
cana-5209	140	18	)	)	PUNCT
cana-5209	140	19	is	be	AUX
cana-5209	140	20	defined	define	VERB
cana-5209	140	21	by	by	ADP
cana-5209	140	22	s	s	NOUN
cana-5209	140	23	=	=	PUNCT
cana-5209	140	24	{	{	PUNCT
cana-5209	140	25	x	x	SYM
cana-5209	140	26	∈	∈	PROPN
cana-5209	140	27	r	r	NOUN
cana-5209	140	28	:	:	PUNCT
cana-5209	140	29	gj(x	gj(x	X
cana-5209	140	30	)	)	PUNCT
cana-5209	140	31	≦	≦	VERB
cana-5209	140	32	0	0	NUM
cana-5209	140	33	}	}	PUNCT
cana-5209	140	34	=	=	PRON
cana-5209	140	35	{	{	PUNCT
cana-5209	140	36	x	x	PUNCT
cana-5209	140	37	∈	∈	NOUN
cana-5209	140	38	r	r	NOUN
cana-5209	140	39	:	:	PUNCT
cana-5209	140	40	x	x	SYM
cana-5209	140	41	≦	≦	NUM
cana-5209	140	42	−1	−1	NOUN
cana-5209	140	43	3	3	NUM
cana-5209	140	44	}	}	PUNCT
cana-5209	140	45	we	we	PRON
cana-5209	140	46	can	can	AUX
cana-5209	140	47	check	check	VERB
cana-5209	140	48	that	that	PRON
cana-5209	140	49	fi(i	fi(i	PUNCT
cana-5209	141	1	=	=	SYM
cana-5209	141	2	1	1	NUM
cana-5209	141	3	,	,	PUNCT
cana-5209	141	4	2	2	NUM
cana-5209	141	5	)	)	PUNCT
cana-5209	141	6	are	be	AUX
cana-5209	141	7	strong	strong	ADJ
cana-5209	141	8	(	(	PUNCT
cana-5209	141	9	c	c	X
cana-5209	141	10	,	,	PUNCT
cana-5209	141	11	αi	αi	PROPN
cana-5209	141	12	,	,	PUNCT
cana-5209	141	13	ηi	ηi	PROPN
cana-5209	141	14	,	,	PUNCT
cana-5209	141	15	ρi	ρi	PROPN
cana-5209	141	16	,	,	PUNCT
cana-5209	141	17	di)-pseudo	di)-pseudo	NOUN
cana-5209	141	18	-	-	PUNCT
cana-5209	141	19	invex	invex	NOUN
cana-5209	141	20	at	at	ADP
cana-5209	141	21	x0	x0	PROPN
cana-5209	141	22	=	=	PUNCT
cana-5209	141	23	−1	−1	NOUN
cana-5209	141	24	3	3	NUM
cana-5209	141	25	,	,	PUNCT
cana-5209	141	26	with	with	ADP
cana-5209	141	27	αi(x	αi(x	NUM
cana-5209	141	28	,	,	PUNCT
cana-5209	141	29	x0	x0	PROPN
cana-5209	141	30	)	)	PUNCT
cana-5209	142	1	=	=	SYM
cana-5209	142	2	1	1	NUM
cana-5209	142	3	,	,	PUNCT
cana-5209	142	4	ηi	ηi	PROPN
cana-5209	142	5	=	=	SYM
cana-5209	142	6	η	η	PROPN
cana-5209	142	7	,	,	PUNCT
cana-5209	142	8	ρi	ρi	NOUN
cana-5209	142	9	=	=	SYM
cana-5209	142	10	0	0	NUM
cana-5209	142	11	and	and	CCONJ
cana-5209	142	12	di(x	di(x	NUM
cana-5209	142	13	,	,	PUNCT
cana-5209	142	14	x0	x0	NUM
cana-5209	142	15	)	)	PUNCT
cana-5209	143	1	=	=	SYM
cana-5209	143	2	1	1	NUM
cana-5209	143	3	,	,	PUNCT
cana-5209	143	4	also	also	ADV
cana-5209	143	5	gj(j	gj(j	NOUN
cana-5209	143	6	∈	∈	PROPN
cana-5209	143	7	i	i	PRON
cana-5209	143	8	)	)	PUNCT
cana-5209	143	9	are	be	AUX
cana-5209	143	10	(	(	PUNCT
cana-5209	143	11	c	c	X
cana-5209	143	12	,	,	PUNCT
cana-5209	143	13	αj	αj	NOUN
cana-5209	143	14	,	,	PUNCT
cana-5209	143	15	ηj	ηj	ADP
cana-5209	143	16	,	,	PUNCT
cana-5209	143	17	ρj	ρj	INTJ
cana-5209	143	18	,	,	PUNCT
cana-5209	143	19	dj)-quasi	dj)-quasi	NOUN
cana-5209	143	20	-	-	PUNCT
cana-5209	143	21	invex	invex	NOUN
cana-5209	143	22	at	at	ADP
cana-5209	143	23	x0	x0	PROPN
cana-5209	143	24	=	=	PUNCT
cana-5209	143	25	−1	−1	NOUN
cana-5209	143	26	3	3	NUM
cana-5209	143	27	,	,	PUNCT
cana-5209	143	28	with	with	ADP
cana-5209	143	29	αj(x	αj(x	NUM
cana-5209	143	30	,	,	PUNCT
cana-5209	143	31	x0	x0	PROPN
cana-5209	143	32	)	)	PUNCT
cana-5209	144	1	=	=	SYM
cana-5209	144	2	1	1	NUM
cana-5209	144	3	,	,	PUNCT
cana-5209	144	4	ηj	ηj	ADP
cana-5209	144	5	=	=	PROPN
cana-5209	144	6	η	η	PROPN
cana-5209	144	7	,	,	PUNCT
cana-5209	144	8	ρj	ρj	NOUN
cana-5209	144	9	=	=	SYM
cana-5209	144	10	0	0	NUM
cana-5209	144	11	and	and	CCONJ
cana-5209	144	12	dj(x	dj(x	NOUN
cana-5209	144	13	,	,	PUNCT
cana-5209	144	14	x0	x0	PROPN
cana-5209	144	15	)	)	PUNCT
cana-5209	144	16	=	=	SYM
cana-5209	145	1	1	1	X
cana-5209	145	2	.	.	PUNCT
cana-5209	145	3	clearly	clearly	ADV
cana-5209	145	4	,	,	PUNCT
cana-5209	145	5	ηi	ηi	PROPN
cana-5209	145	6	=	=	SYM
cana-5209	145	7	ηj	ηj	NOUN
cana-5209	145	8	=	=	PROPN
cana-5209	145	9	η	η	PROPN
cana-5209	145	10	and	and	CCONJ
cana-5209	145	11	x0	x0	PROPN
cana-5209	145	12	=	=	SYM
cana-5209	145	13	−1	−1	NOUN
cana-5209	145	14	3	3	NUM
cana-5209	145	15	is	be	AUX
cana-5209	145	16	a	a	DET
cana-5209	145	17	feasible	feasible	ADJ
cana-5209	145	18	solution	solution	NOUN
cana-5209	145	19	for	for	ADP
cana-5209	145	20	(	(	PUNCT
cana-5209	145	21	sp	sp	ADP
cana-5209	145	22	1	1	NUM
cana-5209	145	23	)	)	PUNCT
cana-5209	145	24	that	that	PRON
cana-5209	145	25	meets	meet	VERB
cana-5209	145	26	the	the	DET
cana-5209	145	27	requirements	requirement	NOUN
cana-5209	145	28	of	of	ADP
cana-5209	145	29	theorem	theorem	NOUN
cana-5209	145	30	4.1	4.1	NUM
cana-5209	145	31	,	,	PUNCT
cana-5209	145	32	in	in	ADP
cana-5209	145	33	which	which	PRON
cana-5209	145	34	u	u	NOUN
cana-5209	145	35	=	=	X
cana-5209	145	36	(	(	PUNCT
cana-5209	145	37	3	3	NUM
cana-5209	145	38	4	4	NUM
cana-5209	145	39	,	,	PUNCT
cana-5209	145	40	1	1	NUM
cana-5209	145	41	)	)	PUNCT
cana-5209	145	42	and	and	CCONJ
cana-5209	145	43	v	v	X
cana-5209	145	44	=	=	SYM
cana-5209	145	45	(	(	PUNCT
cana-5209	145	46	0	0	NUM
cana-5209	145	47	,	,	PUNCT
cana-5209	145	48	0	0	NUM
cana-5209	145	49	,	,	PUNCT
cana-5209	145	50	1	1	NUM
cana-5209	145	51	,	,	PUNCT
cana-5209	145	52	0	0	NUM
cana-5209	145	53	,	,	PUNCT
cana-5209	145	54	...	...	PUNCT
cana-5209	145	55	,	,	PUNCT
cana-5209	145	56	0	0	NUM
cana-5209	145	57	,	,	PUNCT
cana-5209	145	58	...	...	PUNCT
cana-5209	145	59	)	)	PUNCT
cana-5209	145	60	.	.	PUNCT
cana-5209	146	1	for	for	ADP
cana-5209	146	2	x0	x0	PROPN
cana-5209	146	3	=	=	SYM
cana-5209	146	4	−1	−1	NOUN
cana-5209	146	5	3	3	NUM
cana-5209	146	6	,	,	PUNCT
cana-5209	146	7	i	i	PRON
cana-5209	146	8	=	=	PRON
cana-5209	146	9	{	{	PUNCT
cana-5209	146	10	j	j	PROPN
cana-5209	146	11	∈	∈	PROPN
cana-5209	146	12	j	j	PROPN
cana-5209	146	13	:	:	PUNCT
cana-5209	146	14	gj(x0	gj(x0	ADJ
cana-5209	146	15	)	)	PUNCT
cana-5209	146	16	=	=	SYM
cana-5209	146	17	0	0	X
cana-5209	146	18	}	}	PUNCT
cana-5209	146	19	=	=	SYM
cana-5209	146	20	{	{	PUNCT
cana-5209	146	21	3	3	NUM
cana-5209	146	22	}	}	PUNCT
cana-5209	146	23	is	be	AUX
cana-5209	146	24	the	the	DET
cana-5209	146	25	index	index	NOUN
cana-5209	146	26	set	set	NOUN
cana-5209	146	27	of	of	ADP
cana-5209	146	28	active	active	ADJ
cana-5209	146	29	constraints	constraint	NOUN
cana-5209	146	30	.	.	PUNCT
cana-5209	147	1	we	we	PRON
cana-5209	147	2	conclude	conclude	VERB
cana-5209	147	3	that	that	SCONJ
cana-5209	147	4	there	there	PRON
cana-5209	147	5	is	be	VERB
cana-5209	147	6	no	no	DET
cana-5209	147	7	point	point	NOUN
cana-5209	147	8	x	x	X
cana-5209	147	9	∈	∈	NOUN
cana-5209	147	10	s	s	VERB
cana-5209	147	11	such	such	ADJ
cana-5209	147	12	that	that	SCONJ
cana-5209	147	13	f(x	f(x	PROPN
cana-5209	147	14	)	)	PUNCT
cana-5209	147	15	≤	≤	NUM
cana-5209	147	16	f(x0	f(x0	NOUN
cana-5209	147	17	)	)	PUNCT
cana-5209	147	18	.	.	PUNCT
cana-5209	148	1	hence	hence	ADV
cana-5209	148	2	,	,	PUNCT
cana-5209	148	3	x0	x0	PROPN
cana-5209	148	4	=	=	PUNCT
cana-5209	148	5	−1	−1	NOUN
cana-5209	148	6	3	3	NUM
cana-5209	148	7	is	be	AUX
cana-5209	148	8	an	an	DET
cana-5209	148	9	efficient	efficient	ADJ
cana-5209	148	10	solution	solution	NOUN
cana-5209	148	11	of	of	ADP
cana-5209	148	12	(	(	PUNCT
cana-5209	148	13	sp	sp	ADP
cana-5209	148	14	1	1	NUM
cana-5209	148	15	)	)	PUNCT
cana-5209	148	16	.	.	PUNCT
cana-5209	149	1	theorem	theorem	VERB
cana-5209	149	2	4.2	4.2	NUM
cana-5209	149	3	.	.	PUNCT
cana-5209	150	1	if	if	SCONJ
cana-5209	150	2	there	there	PRON
cana-5209	150	3	exist	exist	VERB
cana-5209	150	4	a	a	DET
cana-5209	150	5	feasible	feasible	ADJ
cana-5209	150	6	solution	solution	NOUN
cana-5209	150	7	x0	x0	PROPN
cana-5209	150	8	for	for	ADP
cana-5209	150	9	(	(	PUNCT
cana-5209	150	10	sp	sp	NOUN
cana-5209	150	11	)	)	PUNCT
cana-5209	150	12	and	and	CCONJ
cana-5209	150	13	,	,	PUNCT
cana-5209	150	14	u	u	PROPN
cana-5209	150	15	∈	∈	PROPN
cana-5209	150	16	rp	rp	NOUN
cana-5209	150	17	and	and	CCONJ
cana-5209	150	18	v	v	NOUN
cana-5209	150	19	=	=	PUNCT
cana-5209	150	20	(	(	PUNCT
cana-5209	150	21	vj)j∈j	vj)j∈j	NUM
cana-5209	150	22	are	be	AUX
cana-5209	150	23	vectors	vector	NOUN
cana-5209	150	24	which	which	PRON
cana-5209	150	25	satisfies	satisfy	VERB
cana-5209	150	26	ut∇f(x0	ut∇f(x0	ADJ
cana-5209	150	27	)	)	PUNCT
cana-5209	151	1	+	+	CCONJ
cana-5209	151	2	vti	vti	PROPN
cana-5209	151	3	∇gi(x0	∇gi(x0	PROPN
cana-5209	151	4	)	)	PUNCT
cana-5209	151	5	=	=	SYM
cana-5209	151	6	0	0	NUM
cana-5209	151	7	,	,	PUNCT
cana-5209	151	8	vt	vt	NOUN
cana-5209	151	9	g(x0	g(x0	NOUN
cana-5209	151	10	)	)	PUNCT
cana-5209	151	11	=	=	SYM
cana-5209	151	12	0	0	NUM
cana-5209	151	13	,	,	PUNCT
cana-5209	151	14	u	u	NOUN
cana-5209	151	15	≥	≥	NOUN
cana-5209	151	16	0	0	NUM
cana-5209	151	17	,	,	PUNCT
cana-5209	151	18	v	v	PRON
cana-5209	151	19	≥	≥	NOUN
cana-5209	151	20	0	0	NUM
cana-5209	151	21	and	and	CCONJ
cana-5209	151	22	vj	vj	INTJ
cana-5209	151	23	̸=	̸=	PROPN
cana-5209	151	24	0	0	NUM
cana-5209	151	25	for	for	ADP
cana-5209	151	26	finitely	finitely	ADV
cana-5209	151	27	many	many	ADJ
cana-5209	151	28	j	j	PROPN
cana-5209	151	29	∈	∈	PROPN
cana-5209	151	30	i.	i.	NOUN
cana-5209	151	31	also	also	ADV
cana-5209	151	32	,	,	PUNCT
cana-5209	151	33	if	if	SCONJ
cana-5209	151	34	f	f	PROPN
cana-5209	151	35	is	be	AUX
cana-5209	151	36	weak	weak	ADJ
cana-5209	151	37	strictly	strictly	ADV
cana-5209	151	38	(	(	PUNCT
cana-5209	151	39	c	c	X
cana-5209	151	40	,	,	PUNCT
cana-5209	151	41	α1	α1	PROPN
cana-5209	151	42	,	,	PUNCT
cana-5209	151	43	η1	η1	NOUN
cana-5209	151	44	,	,	PUNCT
cana-5209	151	45	ρ1	ρ1	NOUN
cana-5209	151	46	,	,	PUNCT
cana-5209	151	47	d1)-pseudo	d1)-pseudo	NOUN
cana-5209	151	48	-	-	NOUN
cana-5209	151	49	invex	invex	NOUN
cana-5209	151	50	at	at	ADP
cana-5209	151	51	x0	x0	PROPN
cana-5209	151	52	and	and	CCONJ
cana-5209	151	53	gi	gi	PROPN
cana-5209	151	54	is	be	AUX
cana-5209	151	55	(	(	PUNCT
cana-5209	151	56	c	c	NOUN
cana-5209	151	57	,	,	PUNCT
cana-5209	151	58	α2	α2	ADJ
cana-5209	151	59	,	,	PUNCT
cana-5209	151	60	η2	η2	NOUN
cana-5209	151	61	,	,	PUNCT
cana-5209	151	62	ρ2	ρ2	NOUN
cana-5209	151	63	,	,	PUNCT
cana-5209	151	64	d2)-quasi	d2)-quasi	NOUN
cana-5209	151	65	-	-	NOUN
cana-5209	151	66	invex	invex	NOUN
cana-5209	151	67	at	at	ADP
cana-5209	151	68	x0	x0	PROPN
cana-5209	151	69	with	with	ADP
cana-5209	151	70	respect	respect	NOUN
cana-5209	151	71	to	to	ADP
cana-5209	151	72	same	same	ADJ
cana-5209	151	73	η(x	η(x	NOUN
cana-5209	151	74	,	,	PUNCT
cana-5209	151	75	x0	x0	PROPN
cana-5209	151	76	)	)	PUNCT
cana-5209	151	77	(	(	PUNCT
cana-5209	151	78	i.e.	i.e.	X
cana-5209	151	79	,	,	PUNCT
cana-5209	151	80	η	η	PROPN
cana-5209	151	81	1	1	NUM
cana-5209	151	82	=	=	SYM
cana-5209	151	83	η2	η2	X
cana-5209	151	84	=	=	SYM
cana-5209	151	85	η	η	NOUN
cana-5209	151	86	)	)	PUNCT
cana-5209	151	87	with	with	ADP
cana-5209	151	88	p∑	p∑	PROPN
cana-5209	151	89	i=1	i=1	PRON
cana-5209	151	90	uiρ	uiρ	ADJ
cana-5209	152	1	1	1	NUM
cana-5209	152	2	i	i	PRON
cana-5209	152	3	d1i	d1i	VERB
cana-5209	152	4	(	(	PUNCT
cana-5209	152	5	x	x	X
cana-5209	152	6	,	,	PUNCT
cana-5209	152	7	x0	x0	PROPN
cana-5209	152	8	)	)	PUNCT
cana-5209	152	9	α1	α1	PROPN
cana-5209	153	1	i	i	PRON
cana-5209	153	2	(	(	PUNCT
cana-5209	153	3	x	x	NOUN
cana-5209	153	4	,	,	PUNCT
cana-5209	153	5	x0	x0	PROPN
cana-5209	153	6	)	)	PUNCT
cana-5209	154	1	+	+	CCONJ
cana-5209	154	2	∑	∑	AUX
cana-5209	154	3	j∈i	j∈i	PROPN
cana-5209	154	4	vjρ	vjρ	VERB
cana-5209	154	5	2	2	NUM
cana-5209	154	6	j	j	PROPN
cana-5209	154	7	d2j	d2j	PROPN
cana-5209	154	8	(	(	PUNCT
cana-5209	154	9	x	x	NOUN
cana-5209	154	10	,	,	PUNCT
cana-5209	154	11	x0	x0	PROPN
cana-5209	154	12	)	)	PUNCT
cana-5209	154	13	α2	α2	PROPN
cana-5209	154	14	j	j	PROPN
cana-5209	154	15	(	(	PUNCT
cana-5209	154	16	x	x	PROPN
cana-5209	154	17	,	,	PUNCT
cana-5209	154	18	x0	x0	PROPN
cana-5209	154	19	)	)	PUNCT
cana-5209	154	20	≧	≧	X
cana-5209	155	1	0	0	X
cana-5209	155	2	.	.	PUNCT
cana-5209	156	1	then	then	ADV
cana-5209	156	2	,	,	PUNCT
cana-5209	156	3	x0	x0	PROPN
cana-5209	156	4	is	be	AUX
cana-5209	156	5	an	an	DET
cana-5209	156	6	efficient	efficient	ADJ
cana-5209	156	7	solution	solution	NOUN
cana-5209	156	8	of	of	ADP
cana-5209	156	9	(	(	PUNCT
cana-5209	156	10	sp	sp	NOUN
cana-5209	156	11	)	)	PUNCT
cana-5209	156	12	.	.	PUNCT
cana-5209	157	1	https://internationalpubls.com	https://internationalpubls.com	X
cana-5209	157	2	1144	1144	NUM
cana-5209	157	3	communications	communication	NOUN
cana-5209	157	4	on	on	ADP
cana-5209	157	5	applied	apply	VERB
cana-5209	157	6	nonlinear	nonlinear	ADJ
cana-5209	157	7	analysis	analysis	NOUN
cana-5209	157	8	issn	issn	NOUN
cana-5209	157	9	:	:	PUNCT
cana-5209	157	10	1074	1074	NUM
cana-5209	157	11	-	-	PUNCT
cana-5209	157	12	133x	133x	NUM
cana-5209	157	13	vol	vol	VERB
cana-5209	157	14	32	32	NUM
cana-5209	157	15	no	no	NOUN
cana-5209	157	16	.	.	PUNCT
cana-5209	158	1	10s	10	NOUN
cana-5209	158	2	(	(	PUNCT
cana-5209	158	3	2025	2025	NUM
cana-5209	158	4	)	)	PUNCT
cana-5209	158	5	proof	proof	NOUN
cana-5209	158	6	.	.	PUNCT
cana-5209	159	1	assume	assume	VERB
cana-5209	159	2	that	that	SCONJ
cana-5209	159	3	x0	x0	PROPN
cana-5209	159	4	is	be	AUX
cana-5209	159	5	not	not	PART
cana-5209	159	6	an	an	DET
cana-5209	159	7	efficient	efficient	ADJ
cana-5209	159	8	solution	solution	NOUN
cana-5209	159	9	of	of	ADP
cana-5209	159	10	(	(	PUNCT
cana-5209	159	11	sp	sp	NOUN
cana-5209	159	12	)	)	PUNCT
cana-5209	159	13	.	.	PUNCT
cana-5209	160	1	then	then	ADV
cana-5209	160	2	for	for	ADP
cana-5209	160	3	a	a	DET
cana-5209	160	4	feasible	feasible	ADJ
cana-5209	160	5	solution	solution	NOUN
cana-5209	160	6	x	x	X
cana-5209	160	7	∈	∈	PROPN
cana-5209	160	8	s	s	X
cana-5209	160	9	,	,	PUNCT
cana-5209	160	10	we	we	PRON
cana-5209	160	11	have	have	VERB
cana-5209	160	12	fi(x	fi(x	NUM
cana-5209	160	13	)	)	PUNCT
cana-5209	160	14	≤	≤	NUM
cana-5209	160	15	fi(x0	fi(x0	NOUN
cana-5209	160	16	)	)	PUNCT
cana-5209	160	17	.	.	PUNCT
cana-5209	161	1	since	since	SCONJ
cana-5209	161	2	gi(x0	gi(x0	NOUN
cana-5209	161	3	)	)	PUNCT
cana-5209	161	4	=	=	SYM
cana-5209	161	5	0	0	NUM
cana-5209	161	6	,	,	PUNCT
cana-5209	161	7	hence	hence	ADV
cana-5209	161	8	gi(x	gi(x	PROPN
cana-5209	161	9	)	)	PUNCT
cana-5209	161	10	≦	≦	NUM
cana-5209	161	11	gi(x0	gi(x0	NOUN
cana-5209	161	12	)	)	PUNCT
cana-5209	161	13	.	.	PUNCT
cana-5209	162	1	since	since	SCONJ
cana-5209	162	2	f	f	PROPN
cana-5209	162	3	is	be	AUX
cana-5209	162	4	weak	weak	ADJ
cana-5209	162	5	strictly	strictly	ADV
cana-5209	162	6	(	(	PUNCT
cana-5209	162	7	c	c	X
cana-5209	162	8	,	,	PUNCT
cana-5209	162	9	α1	α1	PROPN
cana-5209	162	10	,	,	PUNCT
cana-5209	162	11	η1	η1	NOUN
cana-5209	162	12	,	,	PUNCT
cana-5209	162	13	ρ1	ρ1	NOUN
cana-5209	162	14	,	,	PUNCT
cana-5209	162	15	d1)-pseudo	d1)-pseudo	NOUN
cana-5209	162	16	-	-	NOUN
cana-5209	162	17	invex	invex	NOUN
cana-5209	162	18	at	at	ADP
cana-5209	162	19	x0	x0	PROPN
cana-5209	162	20	and	and	CCONJ
cana-5209	162	21	gi	gi	PROPN
cana-5209	162	22	is	be	AUX
cana-5209	162	23	(	(	PUNCT
cana-5209	162	24	c	c	NOUN
cana-5209	162	25	,	,	PUNCT
cana-5209	162	26	α2	α2	ADJ
cana-5209	162	27	,	,	PUNCT
cana-5209	162	28	η2	η2	NOUN
cana-5209	162	29	,	,	PUNCT
cana-5209	162	30	ρ2	ρ2	NOUN
cana-5209	162	31	,	,	PUNCT
cana-5209	162	32	d2)-quasi	d2)-quasi	NOUN
cana-5209	162	33	-	-	NOUN
cana-5209	162	34	invex	invex	NOUN
cana-5209	162	35	at	at	ADP
cana-5209	162	36	x0	x0	PROPN
cana-5209	162	37	,	,	PUNCT
cana-5209	162	38	therefore	therefore	ADV
cana-5209	162	39	we	we	PRON
cana-5209	162	40	have	have	VERB
cana-5209	162	41	c(x	c(x	NOUN
cana-5209	162	42	,	,	PUNCT
cana-5209	162	43	x0)(∇fi(x0))η(x	x0)(∇fi(x0))η(x	PROPN
cana-5209	162	44	,	,	PUNCT
cana-5209	162	45	x0	x0	PROPN
cana-5209	162	46	)	)	PUNCT
cana-5209	163	1	+	+	CCONJ
cana-5209	163	2	ρi	ρi	NOUN
cana-5209	163	3	1	1	NUM
cana-5209	163	4	d	d	SYM
cana-5209	163	5	1	1	NUM
cana-5209	163	6	i	i	NOUN
cana-5209	163	7	(	(	PUNCT
cana-5209	163	8	x	x	NOUN
cana-5209	163	9	,	,	PUNCT
cana-5209	163	10	x0	x0	PROPN
cana-5209	163	11	)	)	PUNCT
cana-5209	163	12	α1	α1	PROPN
cana-5209	163	13	i	i	PRON
cana-5209	163	14	(	(	PUNCT
cana-5209	163	15	x	x	NOUN
cana-5209	163	16	,	,	PUNCT
cana-5209	163	17	x0	x0	PROPN
cana-5209	163	18	)	)	PUNCT
cana-5209	163	19	<	<	X
cana-5209	163	20	0	0	NUM
cana-5209	163	21	,	,	PUNCT
cana-5209	163	22	and	and	CCONJ
cana-5209	163	23	c(x	c(x	NOUN
cana-5209	163	24	,	,	PUNCT
cana-5209	163	25	x0)(∇gi(x0))η(x	x0)(∇gi(x0))η(x	PROPN
cana-5209	163	26	,	,	PUNCT
cana-5209	163	27	x0	x0	PROPN
cana-5209	163	28	)	)	PUNCT
cana-5209	164	1	+	+	CCONJ
cana-5209	164	2	ρ2i	ρ2i	ADJ
cana-5209	164	3	d2i(x	d2i(x	PROPN
cana-5209	164	4	,	,	PUNCT
cana-5209	164	5	x0	x0	PROPN
cana-5209	164	6	)	)	PUNCT
cana-5209	164	7	α2	α2	ADJ
cana-5209	164	8	i(x	i(x	PROPN
cana-5209	164	9	,	,	PUNCT
cana-5209	164	10	x0	x0	PROPN
cana-5209	164	11	)	)	PUNCT
cana-5209	164	12	≦	≦	NUM
cana-5209	164	13	0	0	PUNCT
cana-5209	164	14	.	.	PUNCT
cana-5209	165	1	the	the	DET
cana-5209	165	2	remaining	remain	VERB
cana-5209	165	3	part	part	NOUN
cana-5209	165	4	of	of	ADP
cana-5209	165	5	the	the	DET
cana-5209	165	6	proof	proof	NOUN
cana-5209	165	7	is	be	AUX
cana-5209	165	8	similar	similar	ADJ
cana-5209	165	9	to	to	ADP
cana-5209	165	10	proof	proof	NOUN
cana-5209	165	11	of	of	ADP
cana-5209	165	12	theorem	theorem	NOUN
cana-5209	165	13	4.1	4.1	NUM
cana-5209	165	14	.	.	PUNCT
cana-5209	166	1	theorem	theorem	VERB
cana-5209	166	2	4.3	4.3	NUM
cana-5209	166	3	.	.	PUNCT
cana-5209	167	1	if	if	SCONJ
cana-5209	167	2	there	there	PRON
cana-5209	167	3	exist	exist	VERB
cana-5209	167	4	a	a	DET
cana-5209	167	5	feasible	feasible	ADJ
cana-5209	167	6	solution	solution	NOUN
cana-5209	167	7	x0	x0	PROPN
cana-5209	167	8	for	for	ADP
cana-5209	167	9	(	(	PUNCT
cana-5209	167	10	sp	sp	NOUN
cana-5209	167	11	)	)	PUNCT
cana-5209	167	12	and	and	CCONJ
cana-5209	167	13	,	,	PUNCT
cana-5209	167	14	u	u	PROPN
cana-5209	167	15	∈	∈	PROPN
cana-5209	167	16	rp	rp	NOUN
cana-5209	167	17	and	and	CCONJ
cana-5209	167	18	v	v	NOUN
cana-5209	167	19	=	=	PUNCT
cana-5209	167	20	(	(	PUNCT
cana-5209	167	21	vj)j∈j	vj)j∈j	NUM
cana-5209	167	22	are	be	AUX
cana-5209	167	23	vectors	vector	NOUN
cana-5209	167	24	which	which	PRON
cana-5209	167	25	satisfies	satisfy	VERB
cana-5209	167	26	ut∇f(x0	ut∇f(x0	ADJ
cana-5209	167	27	)	)	PUNCT
cana-5209	168	1	+	+	CCONJ
cana-5209	168	2	vti	vti	PROPN
cana-5209	168	3	∇gi(x0	∇gi(x0	PROPN
cana-5209	168	4	)	)	PUNCT
cana-5209	168	5	=	=	SYM
cana-5209	168	6	0	0	NUM
cana-5209	168	7	,	,	PUNCT
cana-5209	168	8	vt	vt	NOUN
cana-5209	168	9	g(x0	g(x0	NOUN
cana-5209	168	10	)	)	PUNCT
cana-5209	168	11	=	=	SYM
cana-5209	168	12	0	0	NUM
cana-5209	168	13	,	,	PUNCT
cana-5209	168	14	u	u	NOUN
cana-5209	168	15	≥	≥	NOUN
cana-5209	168	16	0	0	NUM
cana-5209	168	17	,	,	PUNCT
cana-5209	168	18	v	v	PRON
cana-5209	168	19	≥	≥	NOUN
cana-5209	168	20	0	0	NUM
cana-5209	168	21	and	and	CCONJ
cana-5209	168	22	vj	vj	INTJ
cana-5209	168	23	̸=	̸=	PROPN
cana-5209	168	24	0	0	NUM
cana-5209	168	25	for	for	ADP
cana-5209	168	26	finitely	finitely	ADV
cana-5209	168	27	many	many	ADJ
cana-5209	168	28	j	j	PROPN
cana-5209	168	29	∈	∈	PROPN
cana-5209	168	30	i.	i.	NOUN
cana-5209	168	31	also	also	ADV
cana-5209	168	32	,	,	PUNCT
cana-5209	168	33	if	if	SCONJ
cana-5209	168	34	f	f	PROPN
cana-5209	168	35	is	be	AUX
cana-5209	168	36	weak	weak	ADJ
cana-5209	168	37	(	(	PUNCT
cana-5209	168	38	c	c	NOUN
cana-5209	168	39	,	,	PUNCT
cana-5209	168	40	α1	α1	PROPN
cana-5209	168	41	,	,	PUNCT
cana-5209	168	42	η1	η1	NOUN
cana-5209	168	43	,	,	PUNCT
cana-5209	168	44	ρ1	ρ1	NOUN
cana-5209	168	45	,	,	PUNCT
cana-5209	168	46	d1)-quasi	d1)-quasi	NOUN
cana-5209	168	47	-	-	PUNCT
cana-5209	168	48	invex	invex	NOUN
cana-5209	168	49	at	at	ADP
cana-5209	168	50	x0	x0	PROPN
cana-5209	168	51	and	and	CCONJ
cana-5209	168	52	gi	gi	PROPN
cana-5209	168	53	is	be	AUX
cana-5209	168	54	strictly	strictly	ADV
cana-5209	168	55	(	(	PUNCT
cana-5209	168	56	c	c	X
cana-5209	168	57	,	,	PUNCT
cana-5209	168	58	α2	α2	ADJ
cana-5209	168	59	,	,	PUNCT
cana-5209	168	60	η2	η2	NOUN
cana-5209	168	61	,	,	PUNCT
cana-5209	168	62	ρ2	ρ2	NOUN
cana-5209	168	63	,	,	PUNCT
cana-5209	168	64	d2)-pseudo	d2)-pseudo	NOUN
cana-5209	168	65	-	-	NOUN
cana-5209	168	66	invex	invex	NOUN
cana-5209	168	67	at	at	ADP
cana-5209	168	68	x0	x0	PROPN
cana-5209	168	69	with	with	ADP
cana-5209	168	70	respect	respect	NOUN
cana-5209	168	71	to	to	ADP
cana-5209	168	72	same	same	ADJ
cana-5209	168	73	η(x	η(x	NOUN
cana-5209	168	74	,	,	PUNCT
cana-5209	168	75	x0	x0	PROPN
cana-5209	168	76	)	)	PUNCT
cana-5209	168	77	(	(	PUNCT
cana-5209	168	78	i.e.	i.e.	X
cana-5209	168	79	,	,	PUNCT
cana-5209	168	80	η	η	PROPN
cana-5209	168	81	1	1	NUM
cana-5209	168	82	=	=	SYM
cana-5209	168	83	η2	η2	X
cana-5209	168	84	=	=	SYM
cana-5209	168	85	η	η	NOUN
cana-5209	168	86	)	)	PUNCT
cana-5209	168	87	with	with	ADP
cana-5209	168	88	p∑	p∑	PROPN
cana-5209	169	1	i=1	i=1	PRON
cana-5209	169	2	uiρ	uiρ	ADJ
cana-5209	170	1	1	1	NUM
cana-5209	170	2	i	i	PRON
cana-5209	170	3	d1i	d1i	VERB
cana-5209	170	4	(	(	PUNCT
cana-5209	170	5	x	x	X
cana-5209	170	6	,	,	PUNCT
cana-5209	170	7	x0	x0	PROPN
cana-5209	170	8	)	)	PUNCT
cana-5209	170	9	α1	α1	PROPN
cana-5209	171	1	i	i	PRON
cana-5209	171	2	(	(	PUNCT
cana-5209	171	3	x	x	NOUN
cana-5209	171	4	,	,	PUNCT
cana-5209	171	5	x0	x0	PROPN
cana-5209	171	6	)	)	PUNCT
cana-5209	172	1	+	+	CCONJ
cana-5209	172	2	∑	∑	AUX
cana-5209	172	3	j∈i	j∈i	PROPN
cana-5209	172	4	vjρ	vjρ	VERB
cana-5209	172	5	2	2	NUM
cana-5209	172	6	j	j	PROPN
cana-5209	172	7	d2j	d2j	PROPN
cana-5209	172	8	(	(	PUNCT
cana-5209	172	9	x	x	NOUN
cana-5209	172	10	,	,	PUNCT
cana-5209	172	11	x0	x0	PROPN
cana-5209	172	12	)	)	PUNCT
cana-5209	172	13	α2	α2	PROPN
cana-5209	172	14	j	j	PROPN
cana-5209	172	15	(	(	PUNCT
cana-5209	172	16	x	x	PROPN
cana-5209	172	17	,	,	PUNCT
cana-5209	172	18	x0	x0	PROPN
cana-5209	172	19	)	)	PUNCT
cana-5209	172	20	≧	≧	X
cana-5209	173	1	0	0	X
cana-5209	173	2	.	.	PUNCT
cana-5209	174	1	then	then	ADV
cana-5209	174	2	,	,	PUNCT
cana-5209	174	3	x0	x0	PROPN
cana-5209	174	4	is	be	AUX
cana-5209	174	5	an	an	DET
cana-5209	174	6	efficient	efficient	ADJ
cana-5209	174	7	solution	solution	NOUN
cana-5209	174	8	of	of	ADP
cana-5209	174	9	(	(	PUNCT
cana-5209	174	10	sp	sp	NOUN
cana-5209	174	11	)	)	PUNCT
cana-5209	174	12	.	.	PUNCT
cana-5209	175	1	proof	proof	NOUN
cana-5209	175	2	.	.	PUNCT
cana-5209	176	1	assume	assume	VERB
cana-5209	176	2	that	that	SCONJ
cana-5209	176	3	x0	x0	PROPN
cana-5209	176	4	is	be	AUX
cana-5209	176	5	not	not	PART
cana-5209	176	6	an	an	DET
cana-5209	176	7	efficient	efficient	ADJ
cana-5209	176	8	solution	solution	NOUN
cana-5209	176	9	of	of	ADP
cana-5209	176	10	(	(	PUNCT
cana-5209	176	11	sp	sp	NOUN
cana-5209	176	12	)	)	PUNCT
cana-5209	176	13	.	.	PUNCT
cana-5209	177	1	then	then	ADV
cana-5209	177	2	for	for	ADP
cana-5209	177	3	a	a	DET
cana-5209	177	4	feasible	feasible	ADJ
cana-5209	177	5	solution	solution	NOUN
cana-5209	177	6	x	x	X
cana-5209	177	7	∈	∈	PROPN
cana-5209	177	8	s	s	X
cana-5209	177	9	,	,	PUNCT
cana-5209	177	10	we	we	PRON
cana-5209	177	11	have	have	VERB
cana-5209	177	12	fi(x	fi(x	NUM
cana-5209	177	13	)	)	PUNCT
cana-5209	177	14	≤	≤	NUM
cana-5209	177	15	fi(x0	fi(x0	NOUN
cana-5209	177	16	)	)	PUNCT
cana-5209	177	17	.	.	PUNCT
cana-5209	178	1	since	since	SCONJ
cana-5209	178	2	gi(x0	gi(x0	NOUN
cana-5209	178	3	)	)	PUNCT
cana-5209	178	4	=	=	SYM
cana-5209	178	5	0	0	NUM
cana-5209	178	6	,	,	PUNCT
cana-5209	178	7	hence	hence	ADV
cana-5209	178	8	gi(x	gi(x	PROPN
cana-5209	178	9	)	)	PUNCT
cana-5209	178	10	≦	≦	NUM
cana-5209	178	11	gi(x0	gi(x0	NOUN
cana-5209	178	12	)	)	PUNCT
cana-5209	178	13	.	.	PUNCT
cana-5209	179	1	since	since	SCONJ
cana-5209	179	2	f	f	PROPN
cana-5209	179	3	is	be	AUX
cana-5209	179	4	weak	weak	ADJ
cana-5209	179	5	(	(	PUNCT
cana-5209	179	6	c	c	NOUN
cana-5209	179	7	,	,	PUNCT
cana-5209	179	8	α1	α1	PROPN
cana-5209	179	9	,	,	PUNCT
cana-5209	179	10	η1	η1	NOUN
cana-5209	179	11	,	,	PUNCT
cana-5209	179	12	ρ1	ρ1	NOUN
cana-5209	179	13	,	,	PUNCT
cana-5209	179	14	d1)-quasi	d1)-quasi	NOUN
cana-5209	179	15	-	-	PUNCT
cana-5209	179	16	invex	invex	NOUN
cana-5209	179	17	at	at	ADP
cana-5209	179	18	x0	x0	PROPN
cana-5209	179	19	and	and	CCONJ
cana-5209	179	20	gi	gi	PROPN
cana-5209	179	21	is	be	AUX
cana-5209	179	22	strictly	strictly	ADV
cana-5209	179	23	(	(	PUNCT
cana-5209	179	24	c	c	X
cana-5209	179	25	,	,	PUNCT
cana-5209	179	26	α2	α2	ADJ
cana-5209	179	27	,	,	PUNCT
cana-5209	179	28	η2	η2	NOUN
cana-5209	179	29	,	,	PUNCT
cana-5209	179	30	ρ2	ρ2	NOUN
cana-5209	179	31	,	,	PUNCT
cana-5209	179	32	d2)-pseudo	d2)-pseudo	NOUN
cana-5209	179	33	-	-	NOUN
cana-5209	179	34	invex	invex	NOUN
cana-5209	179	35	at	at	ADP
cana-5209	179	36	x0	x0	PROPN
cana-5209	179	37	,	,	PUNCT
cana-5209	179	38	therefore	therefore	ADV
cana-5209	179	39	we	we	PRON
cana-5209	179	40	have	have	VERB
cana-5209	179	41	c(x	c(x	NOUN
cana-5209	179	42	,	,	PUNCT
cana-5209	179	43	x0)(∇fi(x0))η(x	x0)(∇fi(x0))η(x	PROPN
cana-5209	179	44	,	,	PUNCT
cana-5209	179	45	x0	x0	PROPN
cana-5209	179	46	)	)	PUNCT
cana-5209	180	1	+	+	NUM
cana-5209	180	2	ρ1i	ρ1i	PUNCT
cana-5209	180	3	d1i	d1i	NOUN
cana-5209	180	4	(	(	PUNCT
cana-5209	180	5	x	x	X
cana-5209	180	6	,	,	PUNCT
cana-5209	180	7	x0	x0	PROPN
cana-5209	180	8	)	)	PUNCT
cana-5209	180	9	α1	α1	PROPN
cana-5209	180	10	i	i	PRON
cana-5209	180	11	(	(	PUNCT
cana-5209	180	12	x	x	NOUN
cana-5209	180	13	,	,	PUNCT
cana-5209	180	14	x0	x0	PROPN
cana-5209	180	15	)	)	PUNCT
cana-5209	180	16	≦	≦	NOUN
cana-5209	180	17	0	0	NUM
cana-5209	180	18	,	,	PUNCT
cana-5209	180	19	and	and	CCONJ
cana-5209	180	20	c(x	c(x	NOUN
cana-5209	180	21	,	,	PUNCT
cana-5209	180	22	x0)(∇gi(x0))η(x	x0)(∇gi(x0))η(x	PROPN
cana-5209	180	23	,	,	PUNCT
cana-5209	180	24	x0	x0	PROPN
cana-5209	180	25	)	)	PUNCT
cana-5209	181	1	+	+	CCONJ
cana-5209	181	2	ρ2i	ρ2i	ADJ
cana-5209	181	3	d2i(x	d2i(x	PROPN
cana-5209	181	4	,	,	PUNCT
cana-5209	181	5	x0	x0	PROPN
cana-5209	181	6	)	)	PUNCT
cana-5209	181	7	α2	α2	ADJ
cana-5209	181	8	i(x	i(x	PROPN
cana-5209	181	9	,	,	PUNCT
cana-5209	181	10	x0	x0	PROPN
cana-5209	181	11	)	)	PUNCT
cana-5209	181	12	<	<	X
cana-5209	181	13	0	0	X
cana-5209	181	14	.	.	PUNCT
cana-5209	182	1	the	the	DET
cana-5209	182	2	remaining	remain	VERB
cana-5209	182	3	part	part	NOUN
cana-5209	182	4	of	of	ADP
cana-5209	182	5	the	the	DET
cana-5209	182	6	proof	proof	NOUN
cana-5209	182	7	is	be	AUX
cana-5209	182	8	similar	similar	ADJ
cana-5209	182	9	to	to	ADP
cana-5209	182	10	proof	proof	NOUN
cana-5209	182	11	of	of	ADP
cana-5209	182	12	theorem	theorem	ADJ
cana-5209	182	13	4.1	4.1	NUM
cana-5209	182	14	.	.	PUNCT
cana-5209	183	1	theorem	theorem	VERB
cana-5209	183	2	4.4	4.4	NUM
cana-5209	183	3	.	.	PUNCT
cana-5209	184	1	if	if	SCONJ
cana-5209	184	2	there	there	PRON
cana-5209	184	3	exist	exist	VERB
cana-5209	184	4	a	a	DET
cana-5209	184	5	feasible	feasible	ADJ
cana-5209	184	6	solution	solution	NOUN
cana-5209	184	7	x0	x0	PROPN
cana-5209	184	8	for	for	ADP
cana-5209	184	9	(	(	PUNCT
cana-5209	184	10	sp	sp	NOUN
cana-5209	184	11	)	)	PUNCT
cana-5209	184	12	and	and	CCONJ
cana-5209	184	13	,	,	PUNCT
cana-5209	184	14	u	u	PROPN
cana-5209	184	15	∈	∈	PROPN
cana-5209	184	16	rp	rp	NOUN
cana-5209	184	17	and	and	CCONJ
cana-5209	184	18	v	v	NOUN
cana-5209	184	19	=	=	PUNCT
cana-5209	184	20	(	(	PUNCT
cana-5209	184	21	vj)j∈j	vj)j∈j	NUM
cana-5209	184	22	are	be	AUX
cana-5209	184	23	vectors	vector	NOUN
cana-5209	184	24	which	which	PRON
cana-5209	184	25	satisfies	satisfy	VERB
cana-5209	184	26	ut∇f(x0	ut∇f(x0	ADJ
cana-5209	184	27	)	)	PUNCT
cana-5209	185	1	+	+	CCONJ
cana-5209	185	2	vti	vti	PROPN
cana-5209	185	3	∇gi(x0	∇gi(x0	PROPN
cana-5209	185	4	)	)	PUNCT
cana-5209	185	5	=	=	SYM
cana-5209	185	6	0	0	NUM
cana-5209	185	7	,	,	PUNCT
cana-5209	185	8	vt	vt	NOUN
cana-5209	185	9	g(x0	g(x0	NOUN
cana-5209	185	10	)	)	PUNCT
cana-5209	185	11	=	=	SYM
cana-5209	185	12	0	0	NUM
cana-5209	185	13	,	,	PUNCT
cana-5209	185	14	u	u	NOUN
cana-5209	185	15	≥	≥	NOUN
cana-5209	185	16	0	0	NUM
cana-5209	185	17	,	,	PUNCT
cana-5209	185	18	v	v	PRON
cana-5209	185	19	≥	≥	NOUN
cana-5209	185	20	0	0	NUM
cana-5209	185	21	and	and	CCONJ
cana-5209	185	22	vj	vj	INTJ
cana-5209	185	23	̸=	̸=	PROPN
cana-5209	185	24	0	0	NUM
cana-5209	185	25	for	for	ADP
cana-5209	185	26	finitely	finitely	ADV
cana-5209	185	27	many	many	ADJ
cana-5209	185	28	j	j	PROPN
cana-5209	185	29	∈	∈	PROPN
cana-5209	185	30	i.	i.	NOUN
cana-5209	185	31	also	also	ADV
cana-5209	185	32	,	,	PUNCT
cana-5209	185	33	if	if	SCONJ
cana-5209	185	34	f	f	PROPN
cana-5209	185	35	is	be	AUX
cana-5209	185	36	weak	weak	ADJ
cana-5209	185	37	strictly	strictly	ADV
cana-5209	185	38	(	(	PUNCT
cana-5209	185	39	c	c	X
cana-5209	185	40	,	,	PUNCT
cana-5209	185	41	α1	α1	PROPN
cana-5209	185	42	,	,	PUNCT
cana-5209	185	43	η1	η1	NOUN
cana-5209	185	44	,	,	PUNCT
cana-5209	185	45	ρ1	ρ1	NOUN
cana-5209	185	46	,	,	PUNCT
cana-5209	185	47	d1)-pseudo	d1)-pseudo	NOUN
cana-5209	185	48	-	-	NOUN
cana-5209	185	49	invex	invex	NOUN
cana-5209	185	50	at	at	ADP
cana-5209	185	51	x0	x0	PROPN
cana-5209	185	52	and	and	CCONJ
cana-5209	185	53	gi	gi	PROPN
cana-5209	185	54	is	be	AUX
cana-5209	185	55	strictly	strictly	ADV
cana-5209	185	56	(	(	PUNCT
cana-5209	185	57	c	c	X
cana-5209	185	58	,	,	PUNCT
cana-5209	185	59	α2	α2	ADJ
cana-5209	185	60	,	,	PUNCT
cana-5209	185	61	η2	η2	NOUN
cana-5209	185	62	,	,	PUNCT
cana-5209	185	63	ρ2	ρ2	NOUN
cana-5209	185	64	,	,	PUNCT
cana-5209	185	65	d2)pseudo	d2)pseudo	NOUN
cana-5209	185	66	-	-	NOUN
cana-5209	185	67	invex	invex	NOUN
cana-5209	185	68	at	at	ADP
cana-5209	185	69	x0	x0	PROPN
cana-5209	185	70	with	with	ADP
cana-5209	185	71	respect	respect	NOUN
cana-5209	185	72	to	to	ADP
cana-5209	185	73	same	same	ADJ
cana-5209	185	74	η(x	η(x	NOUN
cana-5209	185	75	,	,	PUNCT
cana-5209	185	76	x0	x0	PROPN
cana-5209	185	77	)	)	PUNCT
cana-5209	185	78	(	(	PUNCT
cana-5209	185	79	i.e.	i.e.	X
cana-5209	185	80	,	,	PUNCT
cana-5209	185	81	η	η	PROPN
cana-5209	185	82	1	1	NUM
cana-5209	185	83	=	=	SYM
cana-5209	185	84	η2	η2	X
cana-5209	185	85	=	=	SYM
cana-5209	185	86	η	η	NOUN
cana-5209	185	87	)	)	PUNCT
cana-5209	185	88	with	with	ADP
cana-5209	185	89	p∑	p∑	PROPN
cana-5209	186	1	i=1	i=1	PRON
cana-5209	186	2	uiρ	uiρ	ADJ
cana-5209	187	1	1	1	NUM
cana-5209	187	2	i	i	PRON
cana-5209	187	3	d1i	d1i	VERB
cana-5209	187	4	(	(	PUNCT
cana-5209	187	5	x	x	X
cana-5209	187	6	,	,	PUNCT
cana-5209	187	7	x0	x0	PROPN
cana-5209	187	8	)	)	PUNCT
cana-5209	187	9	α1	α1	PROPN
cana-5209	188	1	i	i	PRON
cana-5209	188	2	(	(	PUNCT
cana-5209	188	3	x	x	NOUN
cana-5209	188	4	,	,	PUNCT
cana-5209	188	5	x0	x0	PROPN
cana-5209	188	6	)	)	PUNCT
cana-5209	189	1	+	+	CCONJ
cana-5209	189	2	∑	∑	AUX
cana-5209	189	3	j∈i	j∈i	PROPN
cana-5209	189	4	vjρ	vjρ	VERB
cana-5209	189	5	2	2	NUM
cana-5209	189	6	j	j	PROPN
cana-5209	189	7	d2j	d2j	PROPN
cana-5209	189	8	(	(	PUNCT
cana-5209	189	9	x	x	NOUN
cana-5209	189	10	,	,	PUNCT
cana-5209	189	11	x0	x0	PROPN
cana-5209	189	12	)	)	PUNCT
cana-5209	189	13	α2	α2	PROPN
cana-5209	189	14	j	j	PROPN
cana-5209	189	15	(	(	PUNCT
cana-5209	189	16	x	x	PROPN
cana-5209	189	17	,	,	PUNCT
cana-5209	189	18	x0	x0	PROPN
cana-5209	189	19	)	)	PUNCT
cana-5209	189	20	≧	≧	X
cana-5209	190	1	0	0	X
cana-5209	190	2	.	.	PUNCT
cana-5209	191	1	then	then	ADV
cana-5209	191	2	,	,	PUNCT
cana-5209	191	3	x0	x0	PROPN
cana-5209	191	4	is	be	AUX
cana-5209	191	5	an	an	DET
cana-5209	191	6	efficient	efficient	ADJ
cana-5209	191	7	solution	solution	NOUN
cana-5209	191	8	of	of	ADP
cana-5209	191	9	(	(	PUNCT
cana-5209	191	10	sp	sp	NOUN
cana-5209	191	11	)	)	PUNCT
cana-5209	191	12	.	.	PUNCT
cana-5209	192	1	https://internationalpubls.com	https://internationalpubls.com	X
cana-5209	192	2	1145	1145	NUM
cana-5209	192	3	communications	communication	NOUN
cana-5209	192	4	on	on	ADP
cana-5209	192	5	applied	apply	VERB
cana-5209	192	6	nonlinear	nonlinear	ADJ
cana-5209	192	7	analysis	analysis	NOUN
cana-5209	192	8	issn	issn	NOUN
cana-5209	192	9	:	:	PUNCT
cana-5209	192	10	1074	1074	NUM
cana-5209	192	11	-	-	PUNCT
cana-5209	192	12	133x	133x	NUM
cana-5209	192	13	vol	vol	VERB
cana-5209	192	14	32	32	NUM
cana-5209	192	15	no	no	NOUN
cana-5209	192	16	.	.	PUNCT
cana-5209	193	1	10s	10	NOUN
cana-5209	193	2	(	(	PUNCT
cana-5209	193	3	2025	2025	NUM
cana-5209	193	4	)	)	PUNCT
cana-5209	193	5	proof	proof	NOUN
cana-5209	193	6	.	.	PUNCT
cana-5209	194	1	assume	assume	VERB
cana-5209	194	2	that	that	SCONJ
cana-5209	194	3	x0	x0	PROPN
cana-5209	194	4	is	be	AUX
cana-5209	194	5	not	not	PART
cana-5209	194	6	an	an	DET
cana-5209	194	7	efficient	efficient	ADJ
cana-5209	194	8	solution	solution	NOUN
cana-5209	194	9	of	of	ADP
cana-5209	194	10	(	(	PUNCT
cana-5209	194	11	sp	sp	NOUN
cana-5209	194	12	)	)	PUNCT
cana-5209	194	13	.	.	PUNCT
cana-5209	195	1	then	then	ADV
cana-5209	195	2	for	for	ADP
cana-5209	195	3	a	a	DET
cana-5209	195	4	feasible	feasible	ADJ
cana-5209	195	5	solution	solution	NOUN
cana-5209	195	6	x	x	X
cana-5209	195	7	∈	∈	PROPN
cana-5209	195	8	s	s	X
cana-5209	195	9	,	,	PUNCT
cana-5209	195	10	we	we	PRON
cana-5209	195	11	have	have	VERB
cana-5209	195	12	fi(x	fi(x	NUM
cana-5209	195	13	)	)	PUNCT
cana-5209	195	14	≤	≤	NUM
cana-5209	195	15	fi(x0	fi(x0	NOUN
cana-5209	195	16	)	)	PUNCT
cana-5209	195	17	.	.	PUNCT
cana-5209	196	1	since	since	SCONJ
cana-5209	196	2	gi(x0	gi(x0	NOUN
cana-5209	196	3	)	)	PUNCT
cana-5209	196	4	=	=	SYM
cana-5209	196	5	0	0	NUM
cana-5209	196	6	,	,	PUNCT
cana-5209	196	7	hence	hence	ADV
cana-5209	196	8	gi(x	gi(x	PROPN
cana-5209	196	9	)	)	PUNCT
cana-5209	196	10	≦	≦	NUM
cana-5209	196	11	gi(x0	gi(x0	NOUN
cana-5209	196	12	)	)	PUNCT
cana-5209	196	13	.	.	PUNCT
cana-5209	197	1	since	since	SCONJ
cana-5209	197	2	f	f	PROPN
cana-5209	197	3	is	be	AUX
cana-5209	197	4	weak	weak	ADJ
cana-5209	197	5	strictly	strictly	ADV
cana-5209	197	6	(	(	PUNCT
cana-5209	197	7	c	c	X
cana-5209	197	8	,	,	PUNCT
cana-5209	197	9	α1	α1	PROPN
cana-5209	197	10	,	,	PUNCT
cana-5209	197	11	η1	η1	NOUN
cana-5209	197	12	,	,	PUNCT
cana-5209	197	13	ρ1	ρ1	NOUN
cana-5209	197	14	,	,	PUNCT
cana-5209	197	15	d1)-pseudo	d1)-pseudo	NOUN
cana-5209	197	16	-	-	NOUN
cana-5209	197	17	invex	invex	NOUN
cana-5209	197	18	at	at	ADP
cana-5209	197	19	x0	x0	PROPN
cana-5209	197	20	and	and	CCONJ
cana-5209	197	21	gi	gi	PROPN
cana-5209	197	22	is	be	AUX
cana-5209	197	23	strictly	strictly	ADV
cana-5209	197	24	(	(	PUNCT
cana-5209	197	25	c	c	X
cana-5209	197	26	,	,	PUNCT
cana-5209	197	27	α2	α2	ADJ
cana-5209	197	28	,	,	PUNCT
cana-5209	197	29	η2	η2	NOUN
cana-5209	197	30	,	,	PUNCT
cana-5209	197	31	ρ2	ρ2	NOUN
cana-5209	197	32	,	,	PUNCT
cana-5209	197	33	d2)pseudo	d2)pseudo	NOUN
cana-5209	197	34	-	-	NOUN
cana-5209	197	35	invex	invex	NOUN
cana-5209	197	36	at	at	ADP
cana-5209	197	37	x0	x0	PROPN
cana-5209	197	38	,	,	PUNCT
cana-5209	197	39	therefore	therefore	ADV
cana-5209	197	40	we	we	PRON
cana-5209	197	41	have	have	VERB
cana-5209	197	42	c(x	c(x	NOUN
cana-5209	197	43	,	,	PUNCT
cana-5209	197	44	x0)(∇fi(x0))η(x	x0)(∇fi(x0))η(x	PROPN
cana-5209	197	45	,	,	PUNCT
cana-5209	197	46	x0	x0	PROPN
cana-5209	197	47	)	)	PUNCT
cana-5209	198	1	+	+	CCONJ
cana-5209	198	2	ρi	ρi	NOUN
cana-5209	198	3	1	1	NUM
cana-5209	198	4	d	d	SYM
cana-5209	198	5	1	1	NUM
cana-5209	198	6	i	i	NOUN
cana-5209	198	7	(	(	PUNCT
cana-5209	198	8	x	x	NOUN
cana-5209	198	9	,	,	PUNCT
cana-5209	198	10	x0	x0	PROPN
cana-5209	198	11	)	)	PUNCT
cana-5209	198	12	α1	α1	PROPN
cana-5209	198	13	i	i	PRON
cana-5209	198	14	(	(	PUNCT
cana-5209	198	15	x	x	NOUN
cana-5209	198	16	,	,	PUNCT
cana-5209	198	17	x0	x0	PROPN
cana-5209	198	18	)	)	PUNCT
cana-5209	198	19	<	<	X
cana-5209	198	20	0	0	NUM
cana-5209	198	21	,	,	PUNCT
cana-5209	198	22	and	and	CCONJ
cana-5209	198	23	c(x	c(x	NOUN
cana-5209	198	24	,	,	PUNCT
cana-5209	198	25	x0)(∇gi(x0))η(x	x0)(∇gi(x0))η(x	PROPN
cana-5209	198	26	,	,	PUNCT
cana-5209	198	27	x0	x0	PROPN
cana-5209	198	28	)	)	PUNCT
cana-5209	199	1	+	+	CCONJ
cana-5209	199	2	ρ2i	ρ2i	ADJ
cana-5209	199	3	d2i(x	d2i(x	PROPN
cana-5209	199	4	,	,	PUNCT
cana-5209	199	5	x0	x0	PROPN
cana-5209	199	6	)	)	PUNCT
cana-5209	199	7	α2	α2	ADJ
cana-5209	199	8	i(x	i(x	PROPN
cana-5209	199	9	,	,	PUNCT
cana-5209	199	10	x0	x0	PROPN
cana-5209	199	11	)	)	PUNCT
cana-5209	199	12	<	<	X
cana-5209	199	13	0	0	X
cana-5209	199	14	.	.	PUNCT
cana-5209	200	1	the	the	DET
cana-5209	200	2	remaining	remain	VERB
cana-5209	200	3	part	part	NOUN
cana-5209	200	4	of	of	ADP
cana-5209	200	5	the	the	DET
cana-5209	200	6	proof	proof	NOUN
cana-5209	200	7	is	be	AUX
cana-5209	200	8	similar	similar	ADJ
cana-5209	200	9	to	to	ADP
cana-5209	200	10	proof	proof	NOUN
cana-5209	200	11	of	of	ADP
cana-5209	200	12	theorem	theorem	NOUN
cana-5209	200	13	4.1	4.1	NUM
cana-5209	200	14	.	.	PUNCT
cana-5209	201	1	theorem	theorem	VERB
cana-5209	201	2	4.5	4.5	NUM
cana-5209	201	3	.	.	PUNCT
cana-5209	202	1	if	if	SCONJ
cana-5209	202	2	there	there	PRON
cana-5209	202	3	exist	exist	VERB
cana-5209	202	4	a	a	DET
cana-5209	202	5	feasible	feasible	ADJ
cana-5209	202	6	solution	solution	NOUN
cana-5209	202	7	x0	x0	PROPN
cana-5209	202	8	for	for	ADP
cana-5209	202	9	(	(	PUNCT
cana-5209	202	10	sp	sp	NOUN
cana-5209	202	11	)	)	PUNCT
cana-5209	202	12	and	and	CCONJ
cana-5209	202	13	,	,	PUNCT
cana-5209	202	14	u	u	PROPN
cana-5209	202	15	∈	∈	PROPN
cana-5209	202	16	rp	rp	NOUN
cana-5209	202	17	and	and	CCONJ
cana-5209	202	18	v	v	NOUN
cana-5209	202	19	=	=	PUNCT
cana-5209	202	20	(	(	PUNCT
cana-5209	202	21	vj)j∈j	vj)j∈j	NUM
cana-5209	202	22	are	be	AUX
cana-5209	202	23	vectors	vector	NOUN
cana-5209	202	24	which	which	PRON
cana-5209	202	25	satisfies	satisfy	VERB
cana-5209	202	26	ut∇f(x0	ut∇f(x0	ADJ
cana-5209	202	27	)	)	PUNCT
cana-5209	203	1	+	+	CCONJ
cana-5209	203	2	vti	vti	PROPN
cana-5209	203	3	∇gi(x0	∇gi(x0	PROPN
cana-5209	203	4	)	)	PUNCT
cana-5209	203	5	=	=	SYM
cana-5209	203	6	0	0	NUM
cana-5209	203	7	,	,	PUNCT
cana-5209	203	8	vt	vt	NOUN
cana-5209	203	9	g(x0	g(x0	NOUN
cana-5209	203	10	)	)	PUNCT
cana-5209	203	11	=	=	SYM
cana-5209	203	12	0	0	NUM
cana-5209	203	13	,	,	PUNCT
cana-5209	203	14	u	u	NOUN
cana-5209	203	15	≥	≥	NOUN
cana-5209	203	16	0	0	NUM
cana-5209	203	17	,	,	PUNCT
cana-5209	203	18	v	v	PRON
cana-5209	203	19	≥	≥	NOUN
cana-5209	203	20	0	0	NUM
cana-5209	203	21	and	and	CCONJ
cana-5209	203	22	vj	vj	INTJ
cana-5209	203	23	̸=	̸=	PROPN
cana-5209	203	24	0	0	NUM
cana-5209	203	25	for	for	ADP
cana-5209	203	26	finitely	finitely	ADV
cana-5209	203	27	many	many	ADJ
cana-5209	203	28	j	j	PROPN
cana-5209	203	29	∈	∈	PROPN
cana-5209	203	30	i.	i.	NOUN
cana-5209	203	31	also	also	ADV
cana-5209	203	32	,	,	PUNCT
cana-5209	203	33	if	if	SCONJ
cana-5209	203	34	f	f	PROPN
cana-5209	203	35	is	be	AUX
cana-5209	203	36	strong	strong	ADJ
cana-5209	203	37	(	(	PUNCT
cana-5209	203	38	c	c	X
cana-5209	203	39	,	,	PUNCT
cana-5209	203	40	α1	α1	PROPN
cana-5209	203	41	,	,	PUNCT
cana-5209	203	42	η1	η1	NOUN
cana-5209	203	43	,	,	PUNCT
cana-5209	203	44	ρ1	ρ1	NOUN
cana-5209	203	45	,	,	PUNCT
cana-5209	203	46	d1)-pseudo	d1)-pseudo	NOUN
cana-5209	203	47	-	-	NOUN
cana-5209	203	48	invex	invex	NOUN
cana-5209	203	49	at	at	ADP
cana-5209	203	50	x0	x0	PROPN
cana-5209	203	51	and	and	CCONJ
cana-5209	203	52	gi	gi	PROPN
cana-5209	203	53	is	be	AUX
cana-5209	203	54	strictly	strictly	ADV
cana-5209	203	55	(	(	PUNCT
cana-5209	203	56	c	c	X
cana-5209	203	57	,	,	PUNCT
cana-5209	203	58	α2	α2	ADJ
cana-5209	203	59	,	,	PUNCT
cana-5209	203	60	η2	η2	NOUN
cana-5209	203	61	,	,	PUNCT
cana-5209	203	62	ρ2	ρ2	NOUN
cana-5209	203	63	,	,	PUNCT
cana-5209	203	64	d2)-pseudoinvex	d2)-pseudoinvex	VERB
cana-5209	203	65	at	at	ADP
cana-5209	203	66	x0	x0	PROPN
cana-5209	203	67	with	with	ADP
cana-5209	203	68	respect	respect	NOUN
cana-5209	203	69	to	to	ADP
cana-5209	203	70	same	same	ADJ
cana-5209	203	71	η(x	η(x	NOUN
cana-5209	203	72	,	,	PUNCT
cana-5209	203	73	x0	x0	PROPN
cana-5209	203	74	)	)	PUNCT
cana-5209	203	75	(	(	PUNCT
cana-5209	203	76	i.e.	i.e.	X
cana-5209	203	77	,	,	PUNCT
cana-5209	203	78	η	η	PROPN
cana-5209	203	79	1	1	NUM
cana-5209	203	80	=	=	SYM
cana-5209	203	81	η2	η2	X
cana-5209	203	82	=	=	SYM
cana-5209	203	83	η	η	NOUN
cana-5209	203	84	)	)	PUNCT
cana-5209	203	85	with	with	ADP
cana-5209	203	86	p∑	p∑	PROPN
cana-5209	204	1	i=1	i=1	PRON
cana-5209	204	2	uiρ	uiρ	ADJ
cana-5209	205	1	1	1	NUM
cana-5209	205	2	i	i	PRON
cana-5209	205	3	d1i	d1i	VERB
cana-5209	205	4	(	(	PUNCT
cana-5209	205	5	x	x	X
cana-5209	205	6	,	,	PUNCT
cana-5209	205	7	x0	x0	PROPN
cana-5209	205	8	)	)	PUNCT
cana-5209	205	9	α1	α1	PROPN
cana-5209	206	1	i	i	PRON
cana-5209	206	2	(	(	PUNCT
cana-5209	206	3	x	x	NOUN
cana-5209	206	4	,	,	PUNCT
cana-5209	206	5	x0	x0	PROPN
cana-5209	206	6	)	)	PUNCT
cana-5209	207	1	+	+	CCONJ
cana-5209	207	2	∑	∑	AUX
cana-5209	207	3	j∈i	j∈i	PROPN
cana-5209	207	4	vjρ	vjρ	VERB
cana-5209	207	5	2	2	NUM
cana-5209	207	6	j	j	PROPN
cana-5209	207	7	d2j	d2j	PROPN
cana-5209	207	8	(	(	PUNCT
cana-5209	207	9	x	x	NOUN
cana-5209	207	10	,	,	PUNCT
cana-5209	207	11	x0	x0	PROPN
cana-5209	207	12	)	)	PUNCT
cana-5209	207	13	α2	α2	PROPN
cana-5209	207	14	j	j	PROPN
cana-5209	207	15	(	(	PUNCT
cana-5209	207	16	x	x	PROPN
cana-5209	207	17	,	,	PUNCT
cana-5209	207	18	x0	x0	PROPN
cana-5209	207	19	)	)	PUNCT
cana-5209	207	20	≧	≧	X
cana-5209	208	1	0	0	X
cana-5209	208	2	.	.	PUNCT
cana-5209	209	1	then	then	ADV
cana-5209	209	2	,	,	PUNCT
cana-5209	209	3	x0	x0	PROPN
cana-5209	209	4	is	be	AUX
cana-5209	209	5	an	an	DET
cana-5209	209	6	efficient	efficient	ADJ
cana-5209	209	7	solution	solution	NOUN
cana-5209	209	8	of	of	ADP
cana-5209	209	9	(	(	PUNCT
cana-5209	209	10	sp	sp	NOUN
cana-5209	209	11	)	)	PUNCT
cana-5209	209	12	.	.	PUNCT
cana-5209	210	1	proof	proof	NOUN
cana-5209	210	2	.	.	PUNCT
cana-5209	211	1	assume	assume	VERB
cana-5209	211	2	that	that	SCONJ
cana-5209	211	3	x0	x0	PROPN
cana-5209	211	4	is	be	AUX
cana-5209	211	5	not	not	PART
cana-5209	211	6	an	an	DET
cana-5209	211	7	efficient	efficient	ADJ
cana-5209	211	8	solution	solution	NOUN
cana-5209	211	9	of	of	ADP
cana-5209	211	10	(	(	PUNCT
cana-5209	211	11	sp	sp	NOUN
cana-5209	211	12	)	)	PUNCT
cana-5209	211	13	.	.	PUNCT
cana-5209	212	1	then	then	ADV
cana-5209	212	2	for	for	ADP
cana-5209	212	3	a	a	DET
cana-5209	212	4	feasible	feasible	ADJ
cana-5209	212	5	solution	solution	NOUN
cana-5209	212	6	x	x	X
cana-5209	212	7	∈	∈	PROPN
cana-5209	212	8	s	s	X
cana-5209	212	9	,	,	PUNCT
cana-5209	212	10	we	we	PRON
cana-5209	212	11	have	have	VERB
cana-5209	212	12	fi(x	fi(x	NUM
cana-5209	212	13	)	)	PUNCT
cana-5209	212	14	≤	≤	NUM
cana-5209	212	15	fi(x0	fi(x0	NOUN
cana-5209	212	16	)	)	PUNCT
cana-5209	212	17	.	.	PUNCT
cana-5209	213	1	since	since	SCONJ
cana-5209	213	2	gi(x0	gi(x0	NOUN
cana-5209	213	3	)	)	PUNCT
cana-5209	213	4	=	=	SYM
cana-5209	213	5	0	0	NUM
cana-5209	213	6	,	,	PUNCT
cana-5209	213	7	hence	hence	ADV
cana-5209	213	8	gi(x	gi(x	PROPN
cana-5209	213	9	)	)	PUNCT
cana-5209	213	10	≦	≦	NUM
cana-5209	213	11	gi(x0	gi(x0	NOUN
cana-5209	213	12	)	)	PUNCT
cana-5209	213	13	.	.	PUNCT
cana-5209	214	1	since	since	SCONJ
cana-5209	214	2	f	f	PROPN
cana-5209	214	3	is	be	AUX
cana-5209	214	4	strong	strong	ADJ
cana-5209	214	5	(	(	PUNCT
cana-5209	214	6	c	c	X
cana-5209	214	7	,	,	PUNCT
cana-5209	214	8	α1	α1	PROPN
cana-5209	214	9	,	,	PUNCT
cana-5209	214	10	η1	η1	NOUN
cana-5209	214	11	,	,	PUNCT
cana-5209	214	12	ρ1	ρ1	NOUN
cana-5209	214	13	,	,	PUNCT
cana-5209	214	14	d1)-pseudo	d1)-pseudo	NOUN
cana-5209	214	15	-	-	NOUN
cana-5209	214	16	invex	invex	NOUN
cana-5209	214	17	at	at	ADP
cana-5209	214	18	x0	x0	PROPN
cana-5209	214	19	and	and	CCONJ
cana-5209	214	20	gi	gi	PROPN
cana-5209	214	21	is	be	AUX
cana-5209	214	22	strictly	strictly	ADV
cana-5209	214	23	(	(	PUNCT
cana-5209	214	24	c	c	X
cana-5209	214	25	,	,	PUNCT
cana-5209	214	26	α2	α2	ADJ
cana-5209	214	27	,	,	PUNCT
cana-5209	214	28	η2	η2	NOUN
cana-5209	214	29	,	,	PUNCT
cana-5209	214	30	ρ2	ρ2	NOUN
cana-5209	214	31	,	,	PUNCT
cana-5209	214	32	d2)-pseudoinvex	d2)-pseudoinvex	PROPN
cana-5209	214	33	at	at	ADP
cana-5209	214	34	x0	x0	PROPN
cana-5209	214	35	,	,	PUNCT
cana-5209	214	36	therefore	therefore	ADV
cana-5209	214	37	we	we	PRON
cana-5209	214	38	have	have	VERB
cana-5209	214	39	c(x	c(x	NOUN
cana-5209	214	40	,	,	PUNCT
cana-5209	214	41	x0)(∇fi(x0))η(x	x0)(∇fi(x0))η(x	PROPN
cana-5209	214	42	,	,	PUNCT
cana-5209	214	43	x0	x0	PROPN
cana-5209	214	44	)	)	PUNCT
cana-5209	215	1	+	+	NUM
cana-5209	215	2	ρ1i	ρ1i	PUNCT
cana-5209	215	3	d1i	d1i	NOUN
cana-5209	215	4	(	(	PUNCT
cana-5209	215	5	x	x	X
cana-5209	215	6	,	,	PUNCT
cana-5209	215	7	x0	x0	PROPN
cana-5209	215	8	)	)	PUNCT
cana-5209	215	9	α1	α1	PROPN
cana-5209	215	10	i	i	PRON
cana-5209	215	11	(	(	PUNCT
cana-5209	215	12	x	x	NOUN
cana-5209	215	13	,	,	PUNCT
cana-5209	215	14	x0	x0	PROPN
cana-5209	215	15	)	)	PUNCT
cana-5209	215	16	≤	≤	NOUN
cana-5209	215	17	0	0	NUM
cana-5209	215	18	,	,	PUNCT
cana-5209	215	19	and	and	CCONJ
cana-5209	215	20	c(x	c(x	NOUN
cana-5209	215	21	,	,	PUNCT
cana-5209	215	22	x0)(∇gi(x0))η(x	x0)(∇gi(x0))η(x	PROPN
cana-5209	215	23	,	,	PUNCT
cana-5209	215	24	x0	x0	PROPN
cana-5209	215	25	)	)	PUNCT
cana-5209	216	1	+	+	CCONJ
cana-5209	216	2	ρ2i	ρ2i	ADJ
cana-5209	216	3	d2i(x	d2i(x	PROPN
cana-5209	216	4	,	,	PUNCT
cana-5209	216	5	x0	x0	PROPN
cana-5209	216	6	)	)	PUNCT
cana-5209	216	7	α2	α2	ADJ
cana-5209	216	8	i(x	i(x	PROPN
cana-5209	216	9	,	,	PUNCT
cana-5209	216	10	x0	x0	PROPN
cana-5209	216	11	)	)	PUNCT
cana-5209	216	12	<	<	X
cana-5209	216	13	0	0	X
cana-5209	216	14	.	.	PUNCT
cana-5209	217	1	the	the	DET
cana-5209	217	2	remaining	remain	VERB
cana-5209	217	3	part	part	NOUN
cana-5209	217	4	of	of	ADP
cana-5209	217	5	the	the	DET
cana-5209	217	6	proof	proof	NOUN
cana-5209	217	7	is	be	AUX
cana-5209	217	8	similar	similar	ADJ
cana-5209	217	9	to	to	ADP
cana-5209	217	10	proof	proof	NOUN
cana-5209	217	11	of	of	ADP
cana-5209	217	12	theorem	theorem	NOUN
cana-5209	217	13	4.1	4.1	NUM
cana-5209	217	14	.	.	PUNCT
cana-5209	218	1	since	since	SCONJ
cana-5209	218	2	it	it	PRON
cana-5209	218	3	is	be	AUX
cana-5209	218	4	clear	clear	ADJ
cana-5209	218	5	from	from	ADP
cana-5209	218	6	the	the	DET
cana-5209	218	7	definitions	definition	NOUN
cana-5209	218	8	that	that	PRON
cana-5209	218	9	,	,	PUNCT
cana-5209	218	10	an	an	DET
cana-5209	218	11	efficient	efficient	ADJ
cana-5209	218	12	solution	solution	NOUN
cana-5209	218	13	is	be	AUX
cana-5209	218	14	also	also	ADV
cana-5209	218	15	a	a	DET
cana-5209	218	16	weak	weak	ADJ
cana-5209	218	17	efficient	efficient	ADJ
cana-5209	218	18	solution	solution	NOUN
cana-5209	218	19	for	for	ADP
cana-5209	218	20	(	(	PUNCT
cana-5209	218	21	sp	sp	NOUN
cana-5209	218	22	)	)	PUNCT
cana-5209	218	23	but	but	CCONJ
cana-5209	218	24	the	the	DET
cana-5209	218	25	converse	converse	NOUN
cana-5209	218	26	need	need	AUX
cana-5209	218	27	not	not	PART
cana-5209	218	28	be	be	AUX
cana-5209	218	29	true	true	ADJ
cana-5209	218	30	,	,	PUNCT
cana-5209	218	31	therefore	therefore	ADV
cana-5209	218	32	theorem	theorem	VERB
cana-5209	218	33	4.1	4.1	NUM
cana-5209	218	34	theorem	theorem	VERB
cana-5209	218	35	4.5	4.5	NUM
cana-5209	218	36	are	be	AUX
cana-5209	218	37	still	still	ADV
cana-5209	218	38	valid	valid	ADJ
cana-5209	218	39	for	for	ADP
cana-5209	218	40	weak	weak	ADJ
cana-5209	218	41	efficiency	efficiency	NOUN
cana-5209	218	42	.	.	PUNCT
cana-5209	219	1	theorem	theorem	VERB
cana-5209	219	2	4.6	4.6	NUM
cana-5209	219	3	.	.	PUNCT
cana-5209	220	1	if	if	SCONJ
cana-5209	220	2	there	there	PRON
cana-5209	220	3	exist	exist	VERB
cana-5209	220	4	a	a	DET
cana-5209	220	5	feasible	feasible	ADJ
cana-5209	220	6	solution	solution	NOUN
cana-5209	220	7	x0	x0	PROPN
cana-5209	220	8	for	for	ADP
cana-5209	220	9	(	(	PUNCT
cana-5209	220	10	sp	sp	NOUN
cana-5209	220	11	)	)	PUNCT
cana-5209	220	12	and	and	CCONJ
cana-5209	220	13	,	,	PUNCT
cana-5209	220	14	u	u	PROPN
cana-5209	220	15	∈	∈	PROPN
cana-5209	220	16	rp	rp	NOUN
cana-5209	220	17	and	and	CCONJ
cana-5209	220	18	v	v	NOUN
cana-5209	220	19	=	=	PUNCT
cana-5209	220	20	(	(	PUNCT
cana-5209	220	21	vj)j∈j	vj)j∈j	NUM
cana-5209	220	22	are	be	AUX
cana-5209	220	23	vectors	vector	NOUN
cana-5209	220	24	which	which	PRON
cana-5209	220	25	satisfies	satisfy	VERB
cana-5209	220	26	ut∇f(x0	ut∇f(x0	ADJ
cana-5209	220	27	)	)	PUNCT
cana-5209	221	1	+	+	CCONJ
cana-5209	221	2	vti	vti	PROPN
cana-5209	221	3	∇gi(x0	∇gi(x0	PROPN
cana-5209	221	4	)	)	PUNCT
cana-5209	221	5	=	=	SYM
cana-5209	221	6	0	0	NUM
cana-5209	221	7	,	,	PUNCT
cana-5209	221	8	vt	vt	NOUN
cana-5209	221	9	g(x0	g(x0	NOUN
cana-5209	221	10	)	)	PUNCT
cana-5209	221	11	=	=	SYM
cana-5209	221	12	0	0	NUM
cana-5209	221	13	,	,	PUNCT
cana-5209	221	14	u	u	NOUN
cana-5209	221	15	≥	≥	NOUN
cana-5209	221	16	0	0	NUM
cana-5209	221	17	,	,	PUNCT
cana-5209	221	18	v	v	PRON
cana-5209	221	19	≥	≥	NOUN
cana-5209	221	20	0	0	NUM
cana-5209	221	21	and	and	CCONJ
cana-5209	221	22	vj	vj	INTJ
cana-5209	221	23	̸=	̸=	PROPN
cana-5209	221	24	0	0	NUM
cana-5209	221	25	for	for	ADP
cana-5209	221	26	finitely	finitely	ADV
cana-5209	221	27	many	many	ADJ
cana-5209	221	28	j	j	PROPN
cana-5209	221	29	∈	∈	PROPN
cana-5209	221	30	i.	i.	NOUN
cana-5209	221	31	also	also	ADV
cana-5209	221	32	,	,	PUNCT
cana-5209	221	33	if	if	SCONJ
cana-5209	221	34	f	f	PROPN
cana-5209	221	35	is	be	AUX
cana-5209	221	36	(	(	PUNCT
cana-5209	221	37	c	c	X
cana-5209	221	38	,	,	PUNCT
cana-5209	221	39	α1	α1	PROPN
cana-5209	221	40	,	,	PUNCT
cana-5209	221	41	η1	η1	NOUN
cana-5209	221	42	,	,	PUNCT
cana-5209	221	43	ρ1	ρ1	NOUN
cana-5209	221	44	,	,	PUNCT
cana-5209	221	45	d1)-pseudo	d1)-pseudo	NOUN
cana-5209	221	46	-	-	NOUN
cana-5209	221	47	invex	invex	NOUN
cana-5209	221	48	at	at	ADP
cana-5209	221	49	x0	x0	PROPN
cana-5209	221	50	and	and	CCONJ
cana-5209	221	51	gi	gi	PROPN
cana-5209	221	52	is	be	AUX
cana-5209	221	53	strictly	strictly	ADV
cana-5209	221	54	(	(	PUNCT
cana-5209	221	55	c	c	X
cana-5209	221	56	,	,	PUNCT
cana-5209	221	57	α2	α2	ADJ
cana-5209	221	58	,	,	PUNCT
cana-5209	221	59	η2	η2	NOUN
cana-5209	221	60	,	,	PUNCT
cana-5209	221	61	ρ2	ρ2	NOUN
cana-5209	221	62	,	,	PUNCT
cana-5209	221	63	d2)-pseudo	d2)-pseudo	NOUN
cana-5209	221	64	-	-	NOUN
cana-5209	221	65	invex	invex	NOUN
cana-5209	221	66	at	at	ADP
cana-5209	221	67	x0	x0	PROPN
cana-5209	221	68	with	with	ADP
cana-5209	221	69	respect	respect	NOUN
cana-5209	221	70	to	to	ADP
cana-5209	221	71	same	same	ADJ
cana-5209	221	72	η(x	η(x	NOUN
cana-5209	221	73	,	,	PUNCT
cana-5209	221	74	x0	x0	PROPN
cana-5209	221	75	)	)	PUNCT
cana-5209	221	76	(	(	PUNCT
cana-5209	221	77	i.e.	i.e.	X
cana-5209	221	78	,	,	PUNCT
cana-5209	221	79	η	η	PROPN
cana-5209	221	80	1	1	NUM
cana-5209	221	81	=	=	SYM
cana-5209	221	82	η2	η2	X
cana-5209	221	83	=	=	SYM
cana-5209	221	84	η	η	NOUN
cana-5209	221	85	)	)	PUNCT
cana-5209	221	86	with	with	ADP
cana-5209	221	87	p∑	p∑	PROPN
cana-5209	221	88	i=1	i=1	PRON
cana-5209	221	89	uiρ	uiρ	ADJ
cana-5209	222	1	1	1	NUM
cana-5209	222	2	i	i	PRON
cana-5209	222	3	d1i	d1i	VERB
cana-5209	222	4	(	(	PUNCT
cana-5209	222	5	x	x	X
cana-5209	222	6	,	,	PUNCT
cana-5209	222	7	x0	x0	PROPN
cana-5209	222	8	)	)	PUNCT
cana-5209	222	9	α1	α1	PROPN
cana-5209	223	1	i	i	PRON
cana-5209	223	2	(	(	PUNCT
cana-5209	223	3	x	x	NOUN
cana-5209	223	4	,	,	PUNCT
cana-5209	223	5	x0	x0	PROPN
cana-5209	223	6	)	)	PUNCT
cana-5209	224	1	+	+	CCONJ
cana-5209	224	2	∑	∑	AUX
cana-5209	224	3	j∈i	j∈i	PROPN
cana-5209	224	4	vjρ	vjρ	VERB
cana-5209	224	5	2	2	NUM
cana-5209	224	6	j	j	PROPN
cana-5209	224	7	d2j	d2j	PROPN
cana-5209	224	8	(	(	PUNCT
cana-5209	224	9	x	x	NOUN
cana-5209	224	10	,	,	PUNCT
cana-5209	224	11	x0	x0	PROPN
cana-5209	224	12	)	)	PUNCT
cana-5209	224	13	α2	α2	PROPN
cana-5209	224	14	j	j	PROPN
cana-5209	224	15	(	(	PUNCT
cana-5209	224	16	x	x	PROPN
cana-5209	224	17	,	,	PUNCT
cana-5209	224	18	x0	x0	PROPN
cana-5209	224	19	)	)	PUNCT
cana-5209	224	20	≧	≧	X
cana-5209	225	1	0	0	X
cana-5209	225	2	.	.	PUNCT
cana-5209	226	1	then	then	ADV
cana-5209	226	2	,	,	PUNCT
cana-5209	226	3	x0	x0	PROPN
cana-5209	226	4	is	be	AUX
cana-5209	226	5	a	a	DET
cana-5209	226	6	weak	weak	ADJ
cana-5209	226	7	efficient	efficient	ADJ
cana-5209	226	8	solution	solution	NOUN
cana-5209	226	9	of	of	ADP
cana-5209	226	10	(	(	PUNCT
cana-5209	226	11	sp	sp	NOUN
cana-5209	226	12	)	)	PUNCT
cana-5209	226	13	.	.	PUNCT
cana-5209	227	1	https://internationalpubls.com	https://internationalpubls.com	X
cana-5209	227	2	1146	1146	NUM
cana-5209	227	3	communications	communication	NOUN
cana-5209	227	4	on	on	ADP
cana-5209	227	5	applied	apply	VERB
cana-5209	227	6	nonlinear	nonlinear	ADJ
cana-5209	227	7	analysis	analysis	NOUN
cana-5209	227	8	issn	issn	NOUN
cana-5209	227	9	:	:	PUNCT
cana-5209	227	10	1074	1074	NUM
cana-5209	227	11	-	-	PUNCT
cana-5209	227	12	133x	133x	NUM
cana-5209	227	13	vol	vol	VERB
cana-5209	227	14	32	32	NUM
cana-5209	227	15	no	no	NOUN
cana-5209	227	16	.	.	PUNCT
cana-5209	228	1	10s	10	NOUN
cana-5209	228	2	(	(	PUNCT
cana-5209	228	3	2025	2025	NUM
cana-5209	228	4	)	)	PUNCT
cana-5209	228	5	proof	proof	NOUN
cana-5209	228	6	.	.	PUNCT
cana-5209	229	1	assume	assume	VERB
cana-5209	229	2	that	that	SCONJ
cana-5209	229	3	x0	x0	PROPN
cana-5209	229	4	is	be	AUX
cana-5209	229	5	not	not	PART
cana-5209	229	6	a	a	DET
cana-5209	229	7	weak	weak	ADJ
cana-5209	229	8	efficient	efficient	ADJ
cana-5209	229	9	solution	solution	NOUN
cana-5209	229	10	of	of	ADP
cana-5209	229	11	(	(	PUNCT
cana-5209	229	12	sp	sp	NOUN
cana-5209	229	13	)	)	PUNCT
cana-5209	229	14	.	.	PUNCT
cana-5209	230	1	then	then	ADV
cana-5209	230	2	for	for	ADP
cana-5209	230	3	a	a	DET
cana-5209	230	4	feasible	feasible	ADJ
cana-5209	230	5	solution	solution	NOUN
cana-5209	230	6	x	x	X
cana-5209	230	7	∈	∈	PROPN
cana-5209	230	8	s	s	X
cana-5209	230	9	,	,	PUNCT
cana-5209	230	10	we	we	PRON
cana-5209	230	11	have	have	VERB
cana-5209	230	12	fi(x	fi(x	NUM
cana-5209	230	13	)	)	PUNCT
cana-5209	230	14	<	<	X
cana-5209	230	15	fi(x0	fi(x0	PROPN
cana-5209	230	16	)	)	PUNCT
cana-5209	230	17	.	.	PUNCT
cana-5209	231	1	since	since	SCONJ
cana-5209	231	2	gi(x0	gi(x0	NOUN
cana-5209	231	3	)	)	PUNCT
cana-5209	231	4	=	=	SYM
cana-5209	231	5	0	0	NUM
cana-5209	231	6	,	,	PUNCT
cana-5209	231	7	hence	hence	ADV
cana-5209	231	8	gi(x	gi(x	PROPN
cana-5209	231	9	)	)	PUNCT
cana-5209	231	10	≦	≦	NUM
cana-5209	231	11	gi(x0	gi(x0	NOUN
cana-5209	231	12	)	)	PUNCT
cana-5209	231	13	.	.	PUNCT
cana-5209	232	1	since	since	SCONJ
cana-5209	232	2	f	f	PROPN
cana-5209	232	3	is	be	AUX
cana-5209	232	4	(	(	PUNCT
cana-5209	232	5	c	c	X
cana-5209	232	6	,	,	PUNCT
cana-5209	232	7	α1	α1	PROPN
cana-5209	232	8	,	,	PUNCT
cana-5209	232	9	η1	η1	NOUN
cana-5209	232	10	,	,	PUNCT
cana-5209	232	11	ρ1	ρ1	NOUN
cana-5209	232	12	,	,	PUNCT
cana-5209	232	13	d1)-pseudo	d1)-pseudo	NOUN
cana-5209	232	14	-	-	NOUN
cana-5209	232	15	invex	invex	NOUN
cana-5209	232	16	at	at	ADP
cana-5209	232	17	x0	x0	PROPN
cana-5209	232	18	and	and	CCONJ
cana-5209	232	19	gi	gi	PROPN
cana-5209	232	20	is	be	AUX
cana-5209	232	21	strictly	strictly	ADV
cana-5209	232	22	(	(	PUNCT
cana-5209	232	23	c	c	X
cana-5209	232	24	,	,	PUNCT
cana-5209	232	25	α2	α2	ADJ
cana-5209	232	26	,	,	PUNCT
cana-5209	232	27	η2	η2	NOUN
cana-5209	232	28	,	,	PUNCT
cana-5209	232	29	ρ2	ρ2	NOUN
cana-5209	232	30	,	,	PUNCT
cana-5209	232	31	d2)-pseudo	d2)-pseudo	NOUN
cana-5209	232	32	-	-	NOUN
cana-5209	232	33	invex	invex	NOUN
cana-5209	232	34	at	at	ADP
cana-5209	232	35	x0	x0	PROPN
cana-5209	232	36	,	,	PUNCT
cana-5209	232	37	therefore	therefore	ADV
cana-5209	232	38	we	we	PRON
cana-5209	232	39	have	have	VERB
cana-5209	232	40	c(x	c(x	NOUN
cana-5209	232	41	,	,	PUNCT
cana-5209	232	42	x0)(∇fi(x0))η(x	x0)(∇fi(x0))η(x	PROPN
cana-5209	232	43	,	,	PUNCT
cana-5209	232	44	x0	x0	PROPN
cana-5209	232	45	)	)	PUNCT
cana-5209	233	1	+	+	CCONJ
cana-5209	233	2	ρi	ρi	NOUN
cana-5209	233	3	1	1	NUM
cana-5209	233	4	d	d	SYM
cana-5209	233	5	1	1	NUM
cana-5209	233	6	i	i	NOUN
cana-5209	233	7	(	(	PUNCT
cana-5209	233	8	x	x	NOUN
cana-5209	233	9	,	,	PUNCT
cana-5209	233	10	x0	x0	PROPN
cana-5209	233	11	)	)	PUNCT
cana-5209	233	12	α1	α1	PROPN
cana-5209	233	13	i	i	PRON
cana-5209	233	14	(	(	PUNCT
cana-5209	233	15	x	x	NOUN
cana-5209	233	16	,	,	PUNCT
cana-5209	233	17	x0	x0	PROPN
cana-5209	233	18	)	)	PUNCT
cana-5209	233	19	<	<	X
cana-5209	233	20	0	0	NUM
cana-5209	233	21	,	,	PUNCT
cana-5209	233	22	and	and	CCONJ
cana-5209	233	23	c(x	c(x	NOUN
cana-5209	233	24	,	,	PUNCT
cana-5209	233	25	x0)(∇gi(x0))η(x	x0)(∇gi(x0))η(x	PROPN
cana-5209	233	26	,	,	PUNCT
cana-5209	233	27	x0	x0	PROPN
cana-5209	233	28	)	)	PUNCT
cana-5209	234	1	+	+	CCONJ
cana-5209	234	2	ρ2i	ρ2i	ADJ
cana-5209	234	3	d2i(x	d2i(x	PROPN
cana-5209	234	4	,	,	PUNCT
cana-5209	234	5	x0	x0	PROPN
cana-5209	234	6	)	)	PUNCT
cana-5209	234	7	α2	α2	ADJ
cana-5209	234	8	i(x	i(x	PROPN
cana-5209	234	9	,	,	PUNCT
cana-5209	234	10	x0	x0	PROPN
cana-5209	234	11	)	)	PUNCT
cana-5209	234	12	<	<	X
cana-5209	234	13	0	0	X
cana-5209	234	14	.	.	PUNCT
cana-5209	235	1	the	the	DET
cana-5209	235	2	remaining	remain	VERB
cana-5209	235	3	part	part	NOUN
cana-5209	235	4	of	of	ADP
cana-5209	235	5	the	the	DET
cana-5209	235	6	proof	proof	NOUN
cana-5209	235	7	is	be	AUX
cana-5209	235	8	similar	similar	ADJ
cana-5209	235	9	to	to	ADP
cana-5209	235	10	proof	proof	NOUN
cana-5209	235	11	of	of	ADP
cana-5209	235	12	theorem	theorem	NOUN
cana-5209	235	13	4.1	4.1	NUM
cana-5209	235	14	.	.	PUNCT
cana-5209	236	1	theorem	theorem	VERB
cana-5209	236	2	4.7	4.7	NUM
cana-5209	236	3	.	.	PUNCT
cana-5209	237	1	if	if	SCONJ
cana-5209	237	2	there	there	PRON
cana-5209	237	3	exist	exist	VERB
cana-5209	237	4	a	a	DET
cana-5209	237	5	feasible	feasible	ADJ
cana-5209	237	6	solution	solution	NOUN
cana-5209	237	7	x0	x0	PROPN
cana-5209	237	8	for	for	ADP
cana-5209	237	9	(	(	PUNCT
cana-5209	237	10	sp	sp	NOUN
cana-5209	237	11	)	)	PUNCT
cana-5209	237	12	and	and	CCONJ
cana-5209	237	13	,	,	PUNCT
cana-5209	237	14	u	u	PROPN
cana-5209	237	15	∈	∈	PROPN
cana-5209	237	16	rp	rp	NOUN
cana-5209	237	17	and	and	CCONJ
cana-5209	237	18	v	v	NOUN
cana-5209	237	19	=	=	PUNCT
cana-5209	237	20	(	(	PUNCT
cana-5209	237	21	vj)j∈j	vj)j∈j	NUM
cana-5209	237	22	are	be	AUX
cana-5209	237	23	vectors	vector	NOUN
cana-5209	237	24	which	which	PRON
cana-5209	237	25	satisfies	satisfy	VERB
cana-5209	237	26	ut∇f(x0	ut∇f(x0	ADJ
cana-5209	237	27	)	)	PUNCT
cana-5209	238	1	+	+	CCONJ
cana-5209	238	2	vti	vti	PROPN
cana-5209	238	3	∇gi(x0	∇gi(x0	PROPN
cana-5209	238	4	)	)	PUNCT
cana-5209	238	5	=	=	SYM
cana-5209	238	6	0	0	NUM
cana-5209	238	7	,	,	PUNCT
cana-5209	238	8	vt	vt	NOUN
cana-5209	238	9	g(x0	g(x0	NOUN
cana-5209	238	10	)	)	PUNCT
cana-5209	238	11	=	=	SYM
cana-5209	238	12	0	0	NUM
cana-5209	238	13	,	,	PUNCT
cana-5209	238	14	u	u	NOUN
cana-5209	238	15	≥	≥	NOUN
cana-5209	238	16	0	0	NUM
cana-5209	238	17	,	,	PUNCT
cana-5209	238	18	v	v	PRON
cana-5209	238	19	≥	≥	NOUN
cana-5209	238	20	0	0	NUM
cana-5209	238	21	and	and	CCONJ
cana-5209	238	22	vj	vj	INTJ
cana-5209	238	23	̸=	̸=	PROPN
cana-5209	238	24	0	0	NUM
cana-5209	238	25	for	for	ADP
cana-5209	238	26	finitely	finitely	ADV
cana-5209	238	27	many	many	ADJ
cana-5209	238	28	j	j	PROPN
cana-5209	238	29	∈	∈	PROPN
cana-5209	238	30	i.	i.	NOUN
cana-5209	238	31	also	also	ADV
cana-5209	238	32	,	,	PUNCT
cana-5209	238	33	if	if	SCONJ
cana-5209	238	34	f	f	PROPN
cana-5209	238	35	is	be	AUX
cana-5209	238	36	(	(	PUNCT
cana-5209	238	37	c	c	X
cana-5209	238	38	,	,	PUNCT
cana-5209	238	39	α1	α1	PROPN
cana-5209	238	40	,	,	PUNCT
cana-5209	238	41	η1	η1	NOUN
cana-5209	238	42	,	,	PUNCT
cana-5209	238	43	ρ1	ρ1	NOUN
cana-5209	238	44	,	,	PUNCT
cana-5209	238	45	d1)-pseudo	d1)-pseudo	NOUN
cana-5209	238	46	-	-	NOUN
cana-5209	238	47	invex	invex	NOUN
cana-5209	238	48	at	at	ADP
cana-5209	238	49	x0	x0	PROPN
cana-5209	238	50	and	and	CCONJ
cana-5209	238	51	gi	gi	PROPN
cana-5209	238	52	is	be	AUX
cana-5209	238	53	(	(	PUNCT
cana-5209	238	54	c	c	NOUN
cana-5209	238	55	,	,	PUNCT
cana-5209	238	56	α2	α2	ADJ
cana-5209	238	57	,	,	PUNCT
cana-5209	238	58	η2	η2	NOUN
cana-5209	238	59	,	,	PUNCT
cana-5209	238	60	ρ2	ρ2	NOUN
cana-5209	238	61	,	,	PUNCT
cana-5209	238	62	d2)-quasi	d2)-quasi	NOUN
cana-5209	238	63	-	-	NOUN
cana-5209	238	64	invex	invex	NOUN
cana-5209	238	65	at	at	ADP
cana-5209	238	66	x0	x0	PROPN
cana-5209	238	67	with	with	ADP
cana-5209	238	68	respect	respect	NOUN
cana-5209	238	69	to	to	ADP
cana-5209	238	70	same	same	ADJ
cana-5209	238	71	η(x	η(x	NOUN
cana-5209	238	72	,	,	PUNCT
cana-5209	238	73	x0	x0	PROPN
cana-5209	238	74	)	)	PUNCT
cana-5209	238	75	(	(	PUNCT
cana-5209	238	76	i.e.	i.e.	X
cana-5209	238	77	,	,	PUNCT
cana-5209	238	78	η	η	PROPN
cana-5209	238	79	1	1	NUM
cana-5209	238	80	=	=	SYM
cana-5209	238	81	η2	η2	X
cana-5209	238	82	=	=	SYM
cana-5209	238	83	η	η	NOUN
cana-5209	238	84	)	)	PUNCT
cana-5209	238	85	with	with	ADP
cana-5209	238	86	p∑	p∑	PROPN
cana-5209	238	87	i=1	i=1	PRON
cana-5209	238	88	uiρ	uiρ	ADJ
cana-5209	239	1	1	1	NUM
cana-5209	239	2	i	i	PRON
cana-5209	239	3	d1i	d1i	VERB
cana-5209	239	4	(	(	PUNCT
cana-5209	239	5	x	x	X
cana-5209	239	6	,	,	PUNCT
cana-5209	239	7	x0	x0	PROPN
cana-5209	239	8	)	)	PUNCT
cana-5209	239	9	α1	α1	PROPN
cana-5209	240	1	i	i	PRON
cana-5209	240	2	(	(	PUNCT
cana-5209	240	3	x	x	NOUN
cana-5209	240	4	,	,	PUNCT
cana-5209	240	5	x0	x0	PROPN
cana-5209	240	6	)	)	PUNCT
cana-5209	241	1	+	+	CCONJ
cana-5209	241	2	∑	∑	AUX
cana-5209	241	3	j∈i	j∈i	PROPN
cana-5209	241	4	vjρ	vjρ	VERB
cana-5209	241	5	2	2	NUM
cana-5209	241	6	j	j	PROPN
cana-5209	241	7	d2j	d2j	PROPN
cana-5209	241	8	(	(	PUNCT
cana-5209	241	9	x	x	NOUN
cana-5209	241	10	,	,	PUNCT
cana-5209	241	11	x0	x0	PROPN
cana-5209	241	12	)	)	PUNCT
cana-5209	241	13	α2	α2	PROPN
cana-5209	241	14	j	j	PROPN
cana-5209	241	15	(	(	PUNCT
cana-5209	241	16	x	x	PROPN
cana-5209	241	17	,	,	PUNCT
cana-5209	241	18	x0	x0	PROPN
cana-5209	241	19	)	)	PUNCT
cana-5209	241	20	≧	≧	X
cana-5209	242	1	0	0	X
cana-5209	242	2	.	.	PUNCT
cana-5209	243	1	then	then	ADV
cana-5209	243	2	,	,	PUNCT
cana-5209	243	3	x0	x0	PROPN
cana-5209	243	4	is	be	AUX
cana-5209	243	5	a	a	DET
cana-5209	243	6	weak	weak	ADJ
cana-5209	243	7	efficient	efficient	ADJ
cana-5209	243	8	solution	solution	NOUN
cana-5209	243	9	of	of	ADP
cana-5209	243	10	(	(	PUNCT
cana-5209	243	11	sp	sp	NOUN
cana-5209	243	12	)	)	PUNCT
cana-5209	243	13	.	.	PUNCT
cana-5209	244	1	proof	proof	NOUN
cana-5209	244	2	.	.	PUNCT
cana-5209	245	1	assume	assume	VERB
cana-5209	245	2	that	that	SCONJ
cana-5209	245	3	x0	x0	PROPN
cana-5209	245	4	is	be	AUX
cana-5209	245	5	not	not	PART
cana-5209	245	6	a	a	DET
cana-5209	245	7	weak	weak	ADJ
cana-5209	245	8	efficient	efficient	ADJ
cana-5209	245	9	solution	solution	NOUN
cana-5209	245	10	of	of	ADP
cana-5209	245	11	(	(	PUNCT
cana-5209	245	12	sp	sp	NOUN
cana-5209	245	13	)	)	PUNCT
cana-5209	245	14	.	.	PUNCT
cana-5209	246	1	then	then	ADV
cana-5209	246	2	for	for	ADP
cana-5209	246	3	a	a	DET
cana-5209	246	4	feasible	feasible	ADJ
cana-5209	246	5	solution	solution	NOUN
cana-5209	246	6	x	x	X
cana-5209	246	7	∈	∈	PROPN
cana-5209	246	8	s	s	X
cana-5209	246	9	,	,	PUNCT
cana-5209	246	10	we	we	PRON
cana-5209	246	11	have	have	VERB
cana-5209	246	12	fi(x	fi(x	NUM
cana-5209	246	13	)	)	PUNCT
cana-5209	246	14	<	<	X
cana-5209	246	15	fi(x0	fi(x0	PROPN
cana-5209	246	16	)	)	PUNCT
cana-5209	246	17	.	.	PUNCT
cana-5209	247	1	since	since	SCONJ
cana-5209	247	2	gi(x0	gi(x0	NOUN
cana-5209	247	3	)	)	PUNCT
cana-5209	247	4	=	=	SYM
cana-5209	247	5	0	0	NUM
cana-5209	247	6	,	,	PUNCT
cana-5209	247	7	hence	hence	ADV
cana-5209	247	8	gi(x	gi(x	PROPN
cana-5209	247	9	)	)	PUNCT
cana-5209	247	10	≦	≦	NUM
cana-5209	247	11	gi(x0	gi(x0	NOUN
cana-5209	247	12	)	)	PUNCT
cana-5209	247	13	.	.	PUNCT
cana-5209	248	1	since	since	SCONJ
cana-5209	248	2	f	f	PROPN
cana-5209	248	3	is	be	AUX
cana-5209	248	4	(	(	PUNCT
cana-5209	248	5	c	c	X
cana-5209	248	6	,	,	PUNCT
cana-5209	248	7	α1	α1	PROPN
cana-5209	248	8	,	,	PUNCT
cana-5209	248	9	η1	η1	NOUN
cana-5209	248	10	,	,	PUNCT
cana-5209	248	11	ρ1	ρ1	NOUN
cana-5209	248	12	,	,	PUNCT
cana-5209	248	13	d1)-pseudo	d1)-pseudo	NOUN
cana-5209	248	14	-	-	NOUN
cana-5209	248	15	invex	invex	NOUN
cana-5209	248	16	at	at	ADP
cana-5209	248	17	x0	x0	PROPN
cana-5209	248	18	and	and	CCONJ
cana-5209	248	19	gi	gi	PROPN
cana-5209	248	20	is	be	AUX
cana-5209	248	21	(	(	PUNCT
cana-5209	248	22	c	c	X
cana-5209	248	23	,	,	PUNCT
cana-5209	248	24	α	α	PROPN
cana-5209	248	25	2	2	NUM
cana-5209	248	26	,	,	PUNCT
cana-5209	248	27	η2	η2	NOUN
cana-5209	248	28	,	,	PUNCT
cana-5209	248	29	ρ2	ρ2	NOUN
cana-5209	248	30	,	,	PUNCT
cana-5209	248	31	d2)-quasi	d2)-quasi	NOUN
cana-5209	248	32	-	-	NOUN
cana-5209	248	33	invex	invex	NOUN
cana-5209	248	34	at	at	ADP
cana-5209	248	35	x0	x0	PROPN
cana-5209	248	36	,	,	PUNCT
cana-5209	248	37	therefore	therefore	ADV
cana-5209	248	38	we	we	PRON
cana-5209	248	39	have	have	VERB
cana-5209	248	40	c(x	c(x	NOUN
cana-5209	248	41	,	,	PUNCT
cana-5209	248	42	x0)(∇fi(x0))η(x	x0)(∇fi(x0))η(x	PROPN
cana-5209	248	43	,	,	PUNCT
cana-5209	248	44	x0	x0	PROPN
cana-5209	248	45	)	)	PUNCT
cana-5209	249	1	+	+	NUM
cana-5209	249	2	ρ1i	ρ1i	PUNCT
cana-5209	249	3	d1i	d1i	NOUN
cana-5209	249	4	(	(	PUNCT
cana-5209	249	5	x	x	X
cana-5209	249	6	,	,	PUNCT
cana-5209	249	7	x0	x0	PROPN
cana-5209	249	8	)	)	PUNCT
cana-5209	249	9	α1	α1	PROPN
cana-5209	249	10	i	i	PRON
cana-5209	249	11	(	(	PUNCT
cana-5209	249	12	x	x	NOUN
cana-5209	249	13	,	,	PUNCT
cana-5209	249	14	x0	x0	PROPN
cana-5209	249	15	)	)	PUNCT
cana-5209	249	16	<	<	X
cana-5209	249	17	0	0	NUM
cana-5209	249	18	,	,	PUNCT
cana-5209	249	19	and	and	CCONJ
cana-5209	249	20	c(x	c(x	NOUN
cana-5209	249	21	,	,	PUNCT
cana-5209	249	22	x0)(∇gi(x0))η(x	x0)(∇gi(x0))η(x	PROPN
cana-5209	249	23	,	,	PUNCT
cana-5209	249	24	x0	x0	PROPN
cana-5209	249	25	)	)	PUNCT
cana-5209	250	1	+	+	CCONJ
cana-5209	250	2	ρ2i	ρ2i	ADJ
cana-5209	250	3	d2i(x	d2i(x	PROPN
cana-5209	250	4	,	,	PUNCT
cana-5209	250	5	x0	x0	PROPN
cana-5209	250	6	)	)	PUNCT
cana-5209	250	7	α2	α2	ADJ
cana-5209	250	8	i(x	i(x	PROPN
cana-5209	250	9	,	,	PUNCT
cana-5209	250	10	x0	x0	PROPN
cana-5209	250	11	)	)	PUNCT
cana-5209	250	12	≦	≦	NUM
cana-5209	250	13	0	0	PUNCT
cana-5209	250	14	.	.	PUNCT
cana-5209	251	1	the	the	DET
cana-5209	251	2	remaining	remain	VERB
cana-5209	251	3	part	part	NOUN
cana-5209	251	4	of	of	ADP
cana-5209	251	5	the	the	DET
cana-5209	251	6	proof	proof	NOUN
cana-5209	251	7	is	be	AUX
cana-5209	251	8	similar	similar	ADJ
cana-5209	251	9	to	to	ADP
cana-5209	251	10	proof	proof	NOUN
cana-5209	251	11	of	of	ADP
cana-5209	251	12	theorem	theorem	NOUN
cana-5209	251	13	4.1	4.1	NUM
cana-5209	251	14	.	.	NOUN
cana-5209	251	15	5	5	NUM
cana-5209	251	16	duality	duality	NOUN
cana-5209	251	17	we	we	PRON
cana-5209	251	18	now	now	ADV
cana-5209	251	19	prove	prove	VERB
cana-5209	251	20	duality	duality	NOUN
cana-5209	251	21	relations	relation	NOUN
cana-5209	251	22	between	between	ADP
cana-5209	251	23	nonlinear	nonlinear	ADJ
cana-5209	251	24	multiobjective	multiobjective	ADJ
cana-5209	251	25	semiinfinite	semiinfinite	ADJ
cana-5209	251	26	programming	programming	NOUN
cana-5209	251	27	problem	problem	NOUN
cana-5209	251	28	(	(	PUNCT
cana-5209	251	29	sp	sp	NOUN
cana-5209	251	30	)	)	PUNCT
cana-5209	251	31	and	and	CCONJ
cana-5209	251	32	mond	mond	PROPN
cana-5209	251	33	-	-	PUNCT
cana-5209	251	34	weir	weir	NOUN
cana-5209	251	35	-	-	PUNCT
cana-5209	251	36	type	type	NOUN
cana-5209	251	37	dual	dual	ADJ
cana-5209	251	38	problem	problem	NOUN
cana-5209	251	39	(	(	PUNCT
cana-5209	251	40	mwd	mwd	PROPN
cana-5209	251	41	):	):	PUNCT
cana-5209	251	42	(	(	PUNCT
cana-5209	251	43	mwd	mwd	PROPN
cana-5209	251	44	)	)	PUNCT
cana-5209	251	45	max	max	PROPN
cana-5209	251	46	f(y	f(y	PROPN
cana-5209	251	47	)	)	PUNCT
cana-5209	251	48	=	=	PUNCT
cana-5209	251	49	(	(	PUNCT
cana-5209	251	50	f1(y	f1(y	PROPN
cana-5209	251	51	)	)	PUNCT
cana-5209	251	52	,	,	PUNCT
cana-5209	251	53	...	...	PUNCT
cana-5209	251	54	,	,	PUNCT
cana-5209	251	55	fp(y	fp(y	NOUN
cana-5209	251	56	)	)	PUNCT
cana-5209	251	57	)	)	PUNCT
cana-5209	251	58	,	,	PUNCT
cana-5209	251	59	subject	subject	ADJ
cana-5209	251	60	to	to	ADP
cana-5209	251	61	p∑	p∑	PROPN
cana-5209	251	62	i=1	i=1	PROPN
cana-5209	251	63	ui∇fi(y	ui∇fi(y	PROPN
cana-5209	251	64	)	)	PUNCT
cana-5209	252	1	+	+	CCONJ
cana-5209	252	2	∑	∑	PROPN
cana-5209	252	3	j∈j	j∈j	PROPN
cana-5209	252	4	vj∇gj(y	vj∇gj(y	PROPN
cana-5209	252	5	)	)	PUNCT
cana-5209	253	1	=	=	SYM
cana-5209	253	2	0	0	NUM
cana-5209	253	3	,	,	PUNCT
cana-5209	253	4	(	(	PUNCT
cana-5209	253	5	5.1	5.1	NUM
cana-5209	253	6	)	)	PUNCT
cana-5209	253	7	vt	vt	PROPN
cana-5209	253	8	g(y	g(y	PROPN
cana-5209	253	9	)	)	PUNCT
cana-5209	253	10	≧	≧	X
cana-5209	254	1	0	0	NUM
cana-5209	254	2	,	,	PUNCT
cana-5209	254	3	v	v	NOUN
cana-5209	254	4	=	=	SYM
cana-5209	254	5	(	(	PUNCT
cana-5209	254	6	vj)j∈j	vj)j∈j	NUM
cana-5209	254	7	,	,	PUNCT
cana-5209	254	8	vj	vj	PROPN
cana-5209	254	9	∈	∈	PROPN
cana-5209	254	10	r+	r+	NOUN
cana-5209	254	11	and	and	CCONJ
cana-5209	254	12	vj	vj	INTJ
cana-5209	254	13	̸=	̸=	PROPN
cana-5209	254	14	0	0	NUM
cana-5209	254	15	for	for	ADP
cana-5209	254	16	finitely	finitely	ADV
cana-5209	254	17	many	many	ADJ
cana-5209	254	18	j	j	PROPN
cana-5209	254	19	∈	∈	PROPN
cana-5209	254	20	j	j	PROPN
cana-5209	254	21	,	,	PUNCT
cana-5209	254	22	p∑	p∑	AUX
cana-5209	254	23	i=1	i=1	PROPN
cana-5209	254	24	ui	ui	PROPN
cana-5209	255	1	=	=	NOUN
cana-5209	255	2	1	1	NUM
cana-5209	255	3	,	,	PUNCT
cana-5209	255	4	ui	ui	NOUN
cana-5209	255	5	>	>	X
cana-5209	255	6	0	0	PUNCT
cana-5209	256	1	(	(	PUNCT
cana-5209	256	2	i	i	NOUN
cana-5209	256	3	=	=	NOUN
cana-5209	256	4	1	1	NUM
cana-5209	256	5	,	,	PUNCT
cana-5209	256	6	...	...	PUNCT
cana-5209	256	7	,	,	PUNCT
cana-5209	256	8	p	p	X
cana-5209	256	9	)	)	PUNCT
cana-5209	256	10	,	,	PUNCT
cana-5209	256	11	y	y	PROPN
cana-5209	256	12	∈	∈	PROPN
cana-5209	256	13	x	x	PUNCT
cana-5209	256	14	⊆	⊆	NUM
cana-5209	256	15	rn	rn	NOUN
cana-5209	256	16	,	,	PUNCT
cana-5209	256	17	where	where	SCONJ
cana-5209	256	18	fi	fi	NOUN
cana-5209	256	19	and	and	CCONJ
cana-5209	256	20	gi	gi	NOUN
cana-5209	256	21	are	be	AUX
cana-5209	256	22	differnetiable	differnetiable	ADJ
cana-5209	256	23	functions	function	NOUN
cana-5209	256	24	from	from	ADP
cana-5209	256	25	x	x	PUNCT
cana-5209	256	26	to	to	ADP
cana-5209	256	27	r	r	NOUN
cana-5209	256	28	,	,	PUNCT
cana-5209	256	29	and	and	CCONJ
cana-5209	256	30	x	x	X
cana-5209	256	31	is	be	AUX
cana-5209	256	32	nonempty	nonempty	X
cana-5209	256	33	open	open	ADJ
cana-5209	256	34	subset	subset	NOUN
cana-5209	256	35	of	of	ADP
cana-5209	256	36	rn	rn	PROPN
cana-5209	256	37	.	.	PUNCT
cana-5209	257	1	https://internationalpubls.com	https://internationalpubls.com	X
cana-5209	257	2	1147	1147	NUM
cana-5209	257	3	communications	communication	NOUN
cana-5209	257	4	on	on	ADP
cana-5209	257	5	applied	apply	VERB
cana-5209	257	6	nonlinear	nonlinear	ADJ
cana-5209	257	7	analysis	analysis	NOUN
cana-5209	257	8	issn	issn	NOUN
cana-5209	257	9	:	:	PUNCT
cana-5209	257	10	1074	1074	NUM
cana-5209	257	11	-	-	PUNCT
cana-5209	257	12	133x	133x	NUM
cana-5209	257	13	vol	vol	VERB
cana-5209	257	14	32	32	NUM
cana-5209	257	15	no	no	NOUN
cana-5209	257	16	.	.	PUNCT
cana-5209	258	1	10s	10	NOUN
cana-5209	258	2	(	(	PUNCT
cana-5209	258	3	2025	2025	NUM
cana-5209	258	4	)	)	PUNCT
cana-5209	258	5	theorem	theorem	VERB
cana-5209	258	6	5.1	5.1	NUM
cana-5209	258	7	.	.	PUNCT
cana-5209	259	1	(	(	PUNCT
cana-5209	259	2	weak	weak	ADJ
cana-5209	259	3	duality	duality	NOUN
cana-5209	259	4	)	)	PUNCT
cana-5209	259	5	let	let	VERB
cana-5209	259	6	x0	x0	PROPN
cana-5209	259	7	and	and	CCONJ
cana-5209	259	8	(	(	PUNCT
cana-5209	259	9	y0	y0	NOUN
cana-5209	259	10	,	,	PUNCT
cana-5209	259	11	u	u	NOUN
cana-5209	259	12	,	,	PUNCT
cana-5209	259	13	v	v	NOUN
cana-5209	259	14	)	)	PUNCT
cana-5209	259	15	be	be	AUX
cana-5209	259	16	feasible	feasible	ADJ
cana-5209	259	17	solution	solution	NOUN
cana-5209	259	18	for	for	ADP
cana-5209	259	19	(	(	PUNCT
cana-5209	259	20	sp	sp	NOUN
cana-5209	259	21	)	)	PUNCT
cana-5209	259	22	and	and	CCONJ
cana-5209	259	23	(	(	PUNCT
cana-5209	259	24	mwd	mwd	PROPN
cana-5209	259	25	)	)	PUNCT
cana-5209	259	26	respectively	respectively	ADV
cana-5209	259	27	,	,	PUNCT
cana-5209	259	28	and	and	CCONJ
cana-5209	260	1	p∑	p∑	NOUN
cana-5209	261	1	i=1	i=1	PRON
cana-5209	262	1	uiρ	uiρ	ADJ
cana-5209	263	1	1	1	NUM
cana-5209	263	2	i	i	PRON
cana-5209	263	3	d1i	d1i	VERB
cana-5209	263	4	(	(	PUNCT
cana-5209	263	5	x0	x0	PROPN
cana-5209	263	6	,	,	PUNCT
cana-5209	263	7	y0	y0	PROPN
cana-5209	263	8	)	)	PUNCT
cana-5209	263	9	α1	α1	PROPN
cana-5209	263	10	i	i	PRON
cana-5209	263	11	(	(	PUNCT
cana-5209	263	12	x0	x0	PROPN
cana-5209	263	13	,	,	PUNCT
cana-5209	263	14	y0	y0	PROPN
cana-5209	263	15	)	)	PUNCT
cana-5209	264	1	+	+	CCONJ
cana-5209	264	2	∑	∑	PROPN
cana-5209	264	3	j∈j	j∈j	ADJ
cana-5209	264	4	vjρ	vjρ	PROPN
cana-5209	264	5	2	2	NUM
cana-5209	264	6	j	j	PROPN
cana-5209	264	7	d2j	d2j	PROPN
cana-5209	264	8	(	(	PUNCT
cana-5209	264	9	x0	x0	PROPN
cana-5209	264	10	,	,	PUNCT
cana-5209	264	11	y0	y0	PROPN
cana-5209	264	12	)	)	PUNCT
cana-5209	264	13	α2	α2	PROPN
cana-5209	264	14	j	j	PROPN
cana-5209	264	15	(	(	PUNCT
cana-5209	264	16	x0	x0	PROPN
cana-5209	264	17	,	,	PUNCT
cana-5209	264	18	y0	y0	PROPN
cana-5209	264	19	)	)	PUNCT
cana-5209	264	20	≧	≧	X
cana-5209	264	21	0	0	X
cana-5209	264	22	.	.	PUNCT
cana-5209	265	1	(	(	PUNCT
cana-5209	265	2	5.2	5.2	NUM
cana-5209	265	3	)	)	PUNCT
cana-5209	265	4	(	(	PUNCT
cana-5209	265	5	a	a	X
cana-5209	265	6	)	)	PUNCT
cana-5209	265	7	let	let	VERB
cana-5209	265	8	f	f	PRON
cana-5209	265	9	be	be	AUX
cana-5209	265	10	strong	strong	ADJ
cana-5209	265	11	(	(	PUNCT
cana-5209	265	12	c	c	X
cana-5209	265	13	,	,	PUNCT
cana-5209	265	14	α1	α1	PROPN
cana-5209	265	15	,	,	PUNCT
cana-5209	265	16	η1	η1	NOUN
cana-5209	265	17	,	,	PUNCT
cana-5209	265	18	ρ1	ρ1	NOUN
cana-5209	265	19	,	,	PUNCT
cana-5209	265	20	d1)-pseudo	d1)-pseudo	NOUN
cana-5209	265	21	-	-	NOUN
cana-5209	265	22	invex	invex	NOUN
cana-5209	265	23	at	at	ADP
cana-5209	265	24	y0	y0	PROPN
cana-5209	265	25	and	and	CCONJ
cana-5209	265	26	vt	vt	PROPN
cana-5209	265	27	g	g	PROPN
cana-5209	265	28	be	be	AUX
cana-5209	265	29	(	(	PUNCT
cana-5209	265	30	c	c	NOUN
cana-5209	265	31	,	,	PUNCT
cana-5209	265	32	α2	α2	ADJ
cana-5209	265	33	,	,	PUNCT
cana-5209	265	34	η2	η2	NOUN
cana-5209	265	35	,	,	PUNCT
cana-5209	265	36	ρ2	ρ2	NOUN
cana-5209	265	37	,	,	PUNCT
cana-5209	265	38	d2)-quasi	d2)-quasi	NOUN
cana-5209	265	39	-	-	NOUN
cana-5209	265	40	invex	invex	NOUN
cana-5209	265	41	at	at	ADP
cana-5209	265	42	y0	y0	NOUN
cana-5209	265	43	with	with	ADP
cana-5209	265	44	respect	respect	NOUN
cana-5209	265	45	to	to	ADP
cana-5209	265	46	a	a	DET
cana-5209	265	47	common	common	ADJ
cana-5209	265	48	kernal	kernal	ADJ
cana-5209	265	49	η(x	η(x	PROPN
cana-5209	265	50	,	,	PUNCT
cana-5209	265	51	x0	x0	PROPN
cana-5209	265	52	)	)	PUNCT
cana-5209	265	53	(	(	PUNCT
cana-5209	265	54	i.e.	i.e.	X
cana-5209	265	55	,	,	PUNCT
cana-5209	265	56	η	η	PROPN
cana-5209	265	57	1	1	NUM
cana-5209	265	58	=	=	SYM
cana-5209	265	59	η2	η2	X
cana-5209	265	60	=	=	SYM
cana-5209	265	61	η	η	NOUN
cana-5209	265	62	)	)	PUNCT
cana-5209	265	63	.	.	PUNCT
cana-5209	266	1	(	(	PUNCT
cana-5209	266	2	b	b	X
cana-5209	266	3	)	)	PUNCT
cana-5209	266	4	let	let	VERB
cana-5209	266	5	f	f	PRON
cana-5209	266	6	be	be	AUX
cana-5209	266	7	weak	weak	ADJ
cana-5209	266	8	strictly	strictly	ADV
cana-5209	266	9	(	(	PUNCT
cana-5209	266	10	c	c	X
cana-5209	266	11	,	,	PUNCT
cana-5209	266	12	α1	α1	PROPN
cana-5209	266	13	,	,	PUNCT
cana-5209	266	14	η1	η1	NOUN
cana-5209	266	15	,	,	PUNCT
cana-5209	266	16	ρ1	ρ1	NOUN
cana-5209	266	17	,	,	PUNCT
cana-5209	266	18	d1)-pseudo	d1)-pseudo	NOUN
cana-5209	266	19	-	-	NOUN
cana-5209	266	20	invex	invex	NOUN
cana-5209	266	21	at	at	ADP
cana-5209	266	22	y0	y0	PROPN
cana-5209	266	23	and	and	CCONJ
cana-5209	266	24	vt	vt	PROPN
cana-5209	266	25	g	g	PROPN
cana-5209	266	26	be	be	AUX
cana-5209	266	27	(	(	PUNCT
cana-5209	266	28	c	c	NOUN
cana-5209	266	29	,	,	PUNCT
cana-5209	266	30	α2	α2	ADJ
cana-5209	266	31	,	,	PUNCT
cana-5209	266	32	η2	η2	NOUN
cana-5209	266	33	,	,	PUNCT
cana-5209	266	34	ρ2	ρ2	NOUN
cana-5209	266	35	,	,	PUNCT
cana-5209	266	36	d2)-quasiinvex	d2)-quasiinvex	NOUN
cana-5209	266	37	at	at	ADP
cana-5209	266	38	y0	y0	NOUN
cana-5209	266	39	with	with	ADP
cana-5209	266	40	respect	respect	NOUN
cana-5209	266	41	to	to	ADP
cana-5209	266	42	a	a	DET
cana-5209	266	43	common	common	ADJ
cana-5209	266	44	kernal	kernal	ADJ
cana-5209	266	45	η(x	η(x	PROPN
cana-5209	266	46	,	,	PUNCT
cana-5209	266	47	x0	x0	PROPN
cana-5209	266	48	)	)	PUNCT
cana-5209	266	49	(	(	PUNCT
cana-5209	266	50	i.e.	i.e.	X
cana-5209	266	51	,	,	PUNCT
cana-5209	266	52	η	η	PROPN
cana-5209	266	53	1	1	NUM
cana-5209	266	54	=	=	SYM
cana-5209	266	55	η2	η2	X
cana-5209	266	56	=	=	SYM
cana-5209	266	57	η	η	NOUN
cana-5209	266	58	)	)	PUNCT
cana-5209	266	59	.	.	PUNCT
cana-5209	267	1	(	(	PUNCT
cana-5209	267	2	c	c	X
cana-5209	267	3	)	)	PUNCT
cana-5209	267	4	let	let	VERB
cana-5209	267	5	f	f	PRON
cana-5209	267	6	be	be	AUX
cana-5209	267	7	weak	weak	ADJ
cana-5209	267	8	(	(	PUNCT
cana-5209	267	9	c	c	NOUN
cana-5209	267	10	,	,	PUNCT
cana-5209	267	11	α1	α1	PROPN
cana-5209	267	12	,	,	PUNCT
cana-5209	267	13	η1	η1	NOUN
cana-5209	267	14	,	,	PUNCT
cana-5209	267	15	ρ1	ρ1	NOUN
cana-5209	267	16	,	,	PUNCT
cana-5209	267	17	d1)-quasi	d1)-quasi	NOUN
cana-5209	267	18	-	-	PUNCT
cana-5209	267	19	invex	invex	NOUN
cana-5209	267	20	at	at	ADP
cana-5209	267	21	y0	y0	PROPN
cana-5209	267	22	and	and	CCONJ
cana-5209	267	23	vt	vt	PROPN
cana-5209	267	24	g	g	PROPN
cana-5209	267	25	be	be	AUX
cana-5209	267	26	strictly	strictly	ADV
cana-5209	267	27	(	(	PUNCT
cana-5209	267	28	c	c	X
cana-5209	267	29	,	,	PUNCT
cana-5209	267	30	α2	α2	ADJ
cana-5209	267	31	,	,	PUNCT
cana-5209	267	32	η2	η2	NOUN
cana-5209	267	33	,	,	PUNCT
cana-5209	267	34	ρ2	ρ2	NOUN
cana-5209	267	35	,	,	PUNCT
cana-5209	267	36	d2)-pseudoinvex	d2)-pseudoinvex	VERB
cana-5209	267	37	at	at	ADP
cana-5209	267	38	y0	y0	NOUN
cana-5209	267	39	with	with	ADP
cana-5209	267	40	respect	respect	NOUN
cana-5209	267	41	to	to	ADP
cana-5209	267	42	a	a	DET
cana-5209	267	43	common	common	ADJ
cana-5209	267	44	kernal	kernal	ADJ
cana-5209	267	45	η(x	η(x	PROPN
cana-5209	267	46	,	,	PUNCT
cana-5209	267	47	x0	x0	PROPN
cana-5209	267	48	)	)	PUNCT
cana-5209	267	49	(	(	PUNCT
cana-5209	267	50	i.e.	i.e.	X
cana-5209	267	51	,	,	PUNCT
cana-5209	267	52	η	η	PROPN
cana-5209	267	53	1	1	NUM
cana-5209	267	54	=	=	SYM
cana-5209	267	55	η2	η2	X
cana-5209	267	56	=	=	SYM
cana-5209	267	57	η	η	NOUN
cana-5209	267	58	)	)	PUNCT
cana-5209	267	59	.	.	PUNCT
cana-5209	268	1	(	(	PUNCT
cana-5209	268	2	d	d	X
cana-5209	268	3	)	)	PUNCT
cana-5209	268	4	let	let	VERB
cana-5209	268	5	f	f	PRON
cana-5209	268	6	be	be	AUX
cana-5209	268	7	weak	weak	ADJ
cana-5209	268	8	strictly	strictly	ADV
cana-5209	268	9	(	(	PUNCT
cana-5209	268	10	c	c	X
cana-5209	268	11	,	,	PUNCT
cana-5209	268	12	α1	α1	PROPN
cana-5209	268	13	,	,	PUNCT
cana-5209	268	14	η1	η1	NOUN
cana-5209	268	15	,	,	PUNCT
cana-5209	268	16	ρ1	ρ1	NOUN
cana-5209	268	17	,	,	PUNCT
cana-5209	268	18	d1)-pseudo	d1)-pseudo	NOUN
cana-5209	268	19	-	-	NOUN
cana-5209	268	20	invex	invex	NOUN
cana-5209	268	21	at	at	ADP
cana-5209	268	22	y0	y0	PROPN
cana-5209	268	23	and	and	CCONJ
cana-5209	268	24	vt	vt	PROPN
cana-5209	268	25	g	g	PROPN
cana-5209	268	26	be	be	AUX
cana-5209	268	27	strictly	strictly	ADV
cana-5209	268	28	(	(	PUNCT
cana-5209	268	29	c	c	X
cana-5209	268	30	,	,	PUNCT
cana-5209	268	31	α2	α2	ADJ
cana-5209	268	32	,	,	PUNCT
cana-5209	268	33	η2	η2	NOUN
cana-5209	268	34	,	,	PUNCT
cana-5209	268	35	ρ2	ρ2	NOUN
cana-5209	268	36	,	,	PUNCT
cana-5209	268	37	d2)pseudo	d2)pseudo	NOUN
cana-5209	268	38	-	-	NOUN
cana-5209	268	39	invex	invex	NOUN
cana-5209	268	40	at	at	ADP
cana-5209	268	41	y0	y0	NOUN
cana-5209	268	42	with	with	ADP
cana-5209	268	43	respect	respect	NOUN
cana-5209	268	44	to	to	ADP
cana-5209	268	45	a	a	DET
cana-5209	268	46	common	common	ADJ
cana-5209	268	47	kernal	kernal	ADJ
cana-5209	268	48	η(x	η(x	PROPN
cana-5209	268	49	,	,	PUNCT
cana-5209	268	50	x0	x0	PROPN
cana-5209	268	51	)	)	PUNCT
cana-5209	268	52	(	(	PUNCT
cana-5209	268	53	i.e.	i.e.	X
cana-5209	268	54	,	,	PUNCT
cana-5209	268	55	η	η	PROPN
cana-5209	268	56	1	1	NUM
cana-5209	268	57	=	=	SYM
cana-5209	268	58	η2	η2	X
cana-5209	268	59	=	=	SYM
cana-5209	268	60	η	η	NOUN
cana-5209	268	61	)	)	PUNCT
cana-5209	268	62	.	.	PUNCT
cana-5209	269	1	(	(	PUNCT
cana-5209	269	2	e	e	X
cana-5209	269	3	)	)	PUNCT
cana-5209	269	4	let	let	VERB
cana-5209	269	5	f	f	PRON
cana-5209	269	6	be	be	AUX
cana-5209	269	7	strong	strong	ADJ
cana-5209	269	8	(	(	PUNCT
cana-5209	269	9	c	c	X
cana-5209	269	10	,	,	PUNCT
cana-5209	269	11	α1	α1	PROPN
cana-5209	269	12	,	,	PUNCT
cana-5209	269	13	η1	η1	NOUN
cana-5209	269	14	,	,	PUNCT
cana-5209	269	15	ρ1	ρ1	NOUN
cana-5209	269	16	,	,	PUNCT
cana-5209	269	17	d1)-pseudo	d1)-pseudo	NOUN
cana-5209	269	18	-	-	NOUN
cana-5209	269	19	invex	invex	NOUN
cana-5209	269	20	at	at	ADP
cana-5209	269	21	y0	y0	PROPN
cana-5209	269	22	and	and	CCONJ
cana-5209	269	23	vt	vt	PROPN
cana-5209	269	24	g	g	PROPN
cana-5209	269	25	be	be	AUX
cana-5209	269	26	strictly	strictly	ADV
cana-5209	269	27	(	(	PUNCT
cana-5209	269	28	c	c	X
cana-5209	269	29	,	,	PUNCT
cana-5209	269	30	α2	α2	ADJ
cana-5209	269	31	,	,	PUNCT
cana-5209	269	32	η2	η2	NOUN
cana-5209	269	33	,	,	PUNCT
cana-5209	269	34	ρ2	ρ2	NOUN
cana-5209	269	35	,	,	PUNCT
cana-5209	269	36	d2	d2	NOUN
cana-5209	269	37	)	)	PUNCT
cana-5209	269	38	pseudo	pseudo	NOUN
cana-5209	269	39	-	-	NOUN
cana-5209	269	40	invex	invex	NOUN
cana-5209	269	41	at	at	ADP
cana-5209	269	42	y0	y0	NOUN
cana-5209	269	43	with	with	ADP
cana-5209	269	44	respect	respect	NOUN
cana-5209	269	45	to	to	ADP
cana-5209	269	46	a	a	DET
cana-5209	269	47	common	common	ADJ
cana-5209	269	48	kernal	kernal	ADJ
cana-5209	269	49	η(x	η(x	PROPN
cana-5209	269	50	,	,	PUNCT
cana-5209	269	51	x0	x0	PROPN
cana-5209	269	52	)	)	PUNCT
cana-5209	269	53	(	(	PUNCT
cana-5209	269	54	i.e.	i.e.	X
cana-5209	269	55	,	,	PUNCT
cana-5209	269	56	η	η	PROPN
cana-5209	269	57	1	1	NUM
cana-5209	269	58	=	=	SYM
cana-5209	269	59	η2	η2	X
cana-5209	269	60	=	=	SYM
cana-5209	269	61	η	η	NOUN
cana-5209	269	62	)	)	PUNCT
cana-5209	269	63	.	.	PUNCT
cana-5209	270	1	if	if	SCONJ
cana-5209	270	2	any	any	DET
cana-5209	270	3	one	one	NUM
cana-5209	270	4	of	of	ADP
cana-5209	270	5	the	the	DET
cana-5209	270	6	following	follow	VERB
cana-5209	270	7	assumptions	assumption	NOUN
cana-5209	270	8	will	will	AUX
cana-5209	270	9	hold	hold	VERB
cana-5209	270	10	,	,	PUNCT
cana-5209	270	11	then	then	ADV
cana-5209	270	12	the	the	DET
cana-5209	270	13	following	follow	VERB
cana-5209	270	14	inequality	inequality	NOUN
cana-5209	270	15	will	will	AUX
cana-5209	270	16	not	not	PART
cana-5209	270	17	hold	hold	VERB
cana-5209	270	18	f(x0	f(x0	NOUN
cana-5209	270	19	)	)	PUNCT
cana-5209	270	20	≤	≤	NOUN
cana-5209	270	21	f(y0	f(y0	NOUN
cana-5209	270	22	)	)	PUNCT
cana-5209	270	23	.	.	PUNCT
cana-5209	271	1	(	(	PUNCT
cana-5209	271	2	5.3	5.3	NUM
cana-5209	271	3	)	)	PUNCT
cana-5209	271	4	proof	proof	NOUN
cana-5209	271	5	.	.	PUNCT
cana-5209	272	1	(	(	PUNCT
cana-5209	272	2	a	a	X
cana-5209	272	3	)	)	PUNCT
cana-5209	272	4	suppose	suppose	VERB
cana-5209	272	5	that	that	SCONJ
cana-5209	272	6	the	the	DET
cana-5209	272	7	assumption	assumption	NOUN
cana-5209	272	8	(	(	PUNCT
cana-5209	272	9	a	a	X
cana-5209	272	10	)	)	PUNCT
cana-5209	272	11	hold	hold	NOUN
cana-5209	272	12	.	.	PUNCT
cana-5209	273	1	since	since	SCONJ
cana-5209	273	2	x0	x0	PROPN
cana-5209	273	3	and	and	CCONJ
cana-5209	273	4	(	(	PUNCT
cana-5209	273	5	y0	y0	NOUN
cana-5209	273	6	,	,	PUNCT
cana-5209	273	7	u	u	NOUN
cana-5209	273	8	,	,	PUNCT
cana-5209	273	9	v	v	NOUN
cana-5209	273	10	)	)	PUNCT
cana-5209	273	11	are	be	AUX
cana-5209	273	12	the	the	DET
cana-5209	273	13	feasible	feasible	ADJ
cana-5209	273	14	solution	solution	NOUN
cana-5209	273	15	for	for	ADP
cana-5209	273	16	(	(	PUNCT
cana-5209	273	17	sp	sp	NOUN
cana-5209	273	18	)	)	PUNCT
cana-5209	273	19	and	and	CCONJ
cana-5209	273	20	(	(	PUNCT
cana-5209	273	21	mwd	mwd	PROPN
cana-5209	273	22	)	)	PUNCT
cana-5209	273	23	.	.	PUNCT
cana-5209	274	1	therefore	therefore	ADV
cana-5209	274	2	,	,	PUNCT
cana-5209	274	3	vjgj(x0	vjgj(x0	NOUN
cana-5209	274	4	)	)	PUNCT
cana-5209	274	5	≦	≦	NOUN
cana-5209	274	6	0	0	NUM
cana-5209	274	7	≦	≦	NUM
cana-5209	274	8	vjgj(y0	vjgj(y0	NOUN
cana-5209	274	9	)	)	PUNCT
cana-5209	274	10	.	.	PUNCT
cana-5209	275	1	also	also	ADV
cana-5209	275	2	,	,	PUNCT
cana-5209	275	3	vt	vt	PROPN
cana-5209	275	4	g	g	PROPN
cana-5209	275	5	is	be	AUX
cana-5209	275	6	(	(	PUNCT
cana-5209	275	7	c	c	NOUN
cana-5209	275	8	,	,	PUNCT
cana-5209	275	9	α2	α2	ADJ
cana-5209	275	10	,	,	PUNCT
cana-5209	275	11	η2	η2	NOUN
cana-5209	275	12	,	,	PUNCT
cana-5209	275	13	ρ2	ρ2	NOUN
cana-5209	275	14	,	,	PUNCT
cana-5209	275	15	d2)-quasi	d2)-quasi	NOUN
cana-5209	275	16	-	-	NOUN
cana-5209	275	17	invex	invex	NOUN
cana-5209	275	18	at	at	ADP
cana-5209	275	19	y0	y0	PROPN
cana-5209	275	20	,	,	PUNCT
cana-5209	275	21	which	which	PRON
cana-5209	275	22	implies	imply	VERB
cana-5209	275	23	that	that	SCONJ
cana-5209	275	24	c(x0,y0)(vj∇gj(y0))η(x0	c(x0,y0)(vj∇gj(y0))η(x0	NOUN
cana-5209	275	25	,	,	PUNCT
cana-5209	275	26	y0	y0	PROPN
cana-5209	275	27	)	)	PUNCT
cana-5209	275	28	+	+	CCONJ
cana-5209	275	29	ρ2j	ρ2j	NUM
cana-5209	275	30	d2j	d2j	NOUN
cana-5209	275	31	(	(	PUNCT
cana-5209	275	32	x0	x0	PROPN
cana-5209	275	33	,	,	PUNCT
cana-5209	275	34	y0	y0	PROPN
cana-5209	275	35	)	)	PUNCT
cana-5209	275	36	α2	α2	PROPN
cana-5209	275	37	j	j	PROPN
cana-5209	275	38	(	(	PUNCT
cana-5209	275	39	x0	x0	PROPN
cana-5209	275	40	,	,	PUNCT
cana-5209	275	41	y0	y0	PROPN
cana-5209	275	42	)	)	PUNCT
cana-5209	275	43	≦	≦	NOUN
cana-5209	275	44	0	0	NUM
cana-5209	275	45	.	.	PUNCT
cana-5209	276	1	(	(	PUNCT
cana-5209	276	2	5.4	5.4	NUM
cana-5209	276	3	)	)	PUNCT
cana-5209	276	4	let	let	VERB
cana-5209	276	5	(	(	PUNCT
cana-5209	276	6	5.3	5.3	NUM
cana-5209	276	7	)	)	PUNCT
cana-5209	276	8	holds	hold	VERB
cana-5209	276	9	,	,	PUNCT
cana-5209	276	10	then	then	ADV
cana-5209	276	11	by	by	ADP
cana-5209	276	12	the	the	DET
cana-5209	276	13	definition	definition	NOUN
cana-5209	276	14	of	of	ADP
cana-5209	276	15	strong	strong	ADJ
cana-5209	276	16	(	(	PUNCT
cana-5209	276	17	c	c	NOUN
cana-5209	276	18	,	,	PUNCT
cana-5209	276	19	α1	α1	PROPN
cana-5209	276	20	,	,	PUNCT
cana-5209	276	21	η1	η1	NOUN
cana-5209	276	22	,	,	PUNCT
cana-5209	276	23	ρ1	ρ1	NOUN
cana-5209	276	24	,	,	PUNCT
cana-5209	276	25	d1)-pseudo	d1)-pseudo	NOUN
cana-5209	276	26	-	-	NOUN
cana-5209	276	27	invexity	invexity	NOUN
cana-5209	276	28	,	,	PUNCT
cana-5209	276	29	we	we	PRON
cana-5209	276	30	have	have	VERB
cana-5209	276	31	c(x0,y0)(∇fi(y0))η(x0	c(x0,y0)(∇fi(y0))η(x0	PROPN
cana-5209	276	32	,	,	PUNCT
cana-5209	276	33	y0	y0	PROPN
cana-5209	276	34	)	)	PUNCT
cana-5209	277	1	+	+	SYM
cana-5209	277	2	ρ1i	ρ1i	PUNCT
cana-5209	277	3	d1i	d1i	NOUN
cana-5209	277	4	(	(	PUNCT
cana-5209	277	5	x0	x0	PROPN
cana-5209	277	6	,	,	PUNCT
cana-5209	277	7	y0	y0	PROPN
cana-5209	277	8	)	)	PUNCT
cana-5209	277	9	α1	α1	PROPN
cana-5209	277	10	i	i	PRON
cana-5209	277	11	(	(	PUNCT
cana-5209	277	12	x0	x0	PROPN
cana-5209	277	13	,	,	PUNCT
cana-5209	277	14	y0	y0	PROPN
cana-5209	277	15	)	)	PUNCT
cana-5209	277	16	≤	≤	NOUN
cana-5209	277	17	0	0	NUM
cana-5209	277	18	.	.	PUNCT
cana-5209	277	19	(	(	PUNCT
cana-5209	277	20	5.5	5.5	NUM
cana-5209	277	21	)	)	PUNCT
cana-5209	277	22	let	let	VERB
cana-5209	277	23	τ	τ	X
cana-5209	277	24	=	=	PUNCT
cana-5209	277	25	p∑	p∑	X
cana-5209	277	26	i=1	i=1	X
cana-5209	277	27	ui	ui	PROPN
cana-5209	278	1	+	+	CCONJ
cana-5209	278	2	∑	∑	PROPN
cana-5209	278	3	j∈j	j∈j	ADJ
cana-5209	278	4	vj	vj	PROPN
cana-5209	278	5	.	.	PUNCT
cana-5209	279	1	then	then	ADV
cana-5209	279	2	,	,	PUNCT
cana-5209	279	3	by	by	ADP
cana-5209	279	4	the	the	DET
cana-5209	279	5	convexity	convexity	NOUN
cana-5209	279	6	of	of	ADP
cana-5209	279	7	c(x0,y0	c(x0,y0	NOUN
cana-5209	279	8	)	)	PUNCT
cana-5209	279	9	(	(	PUNCT
cana-5209	279	10	·	·	PUNCT
cana-5209	279	11	)	)	PUNCT
cana-5209	279	12	and	and	CCONJ
cana-5209	279	13	equations	equation	NOUN
cana-5209	279	14	(	(	PUNCT
cana-5209	279	15	5.1),(5.2),(5.4),(5.5	5.1),(5.2),(5.4),(5.5	NUM
cana-5209	279	16	)	)	PUNCT
cana-5209	279	17	,	,	PUNCT
cana-5209	279	18	we	we	PRON
cana-5209	279	19	get	get	VERB
cana-5209	279	20	0	0	PUNCT
cana-5209	279	21	>	>	X
cana-5209	279	22	p∑	p∑	X
cana-5209	280	1	i=1	i=1	INTJ
cana-5209	280	2	ui	ui	PROPN
cana-5209	281	1	τ	τ	PROPN
cana-5209	281	2	c(x0,y0)(∇fi(y0))η(x0	c(x0,y0)(∇fi(y0))η(x0	PROPN
cana-5209	281	3	,	,	PUNCT
cana-5209	281	4	y0	y0	PROPN
cana-5209	281	5	)	)	PUNCT
cana-5209	281	6	+	+	CCONJ
cana-5209	281	7	∑	∑	PROPN
cana-5209	281	8	j∈j	j∈j	PROPN
cana-5209	281	9	vj	vj	PROPN
cana-5209	281	10	τ	τ	PROPN
cana-5209	281	11	c(x0,y0)(∇gj(y0))η(x0	c(x0,y0)(∇gj(y0))η(x0	PROPN
cana-5209	281	12	,	,	PUNCT
cana-5209	281	13	y0	y0	PROPN
cana-5209	281	14	)	)	PUNCT
cana-5209	281	15	+	+	NUM
cana-5209	281	16	p∑	p∑	X
cana-5209	282	1	i=1	i=1	PROPN
cana-5209	282	2	ui	ui	PROPN
cana-5209	283	1	τ	τ	PROPN
cana-5209	283	2	ρ1i	ρ1i	PUNCT
cana-5209	283	3	d1i	d1i	PROPN
cana-5209	283	4	(	(	PUNCT
cana-5209	283	5	x0	x0	PROPN
cana-5209	283	6	,	,	PUNCT
cana-5209	283	7	y0	y0	PROPN
cana-5209	283	8	)	)	PUNCT
cana-5209	283	9	α1	α1	PROPN
cana-5209	283	10	i	i	PRON
cana-5209	283	11	(	(	PUNCT
cana-5209	283	12	x0	x0	PROPN
cana-5209	283	13	,	,	PUNCT
cana-5209	283	14	y0	y0	PROPN
cana-5209	283	15	)	)	PUNCT
cana-5209	284	1	+	+	CCONJ
cana-5209	284	2	∑	∑	PROPN
cana-5209	284	3	j∈j	j∈j	PROPN
cana-5209	284	4	vj	vj	PROPN
cana-5209	284	5	τ	τ	PROPN
cana-5209	284	6	ρ2j	ρ2j	X
cana-5209	284	7	d2j	d2j	PROPN
cana-5209	284	8	(	(	PUNCT
cana-5209	284	9	x0	x0	PROPN
cana-5209	284	10	,	,	PUNCT
cana-5209	284	11	y0	y0	PROPN
cana-5209	284	12	)	)	PUNCT
cana-5209	284	13	α2	α2	PROPN
cana-5209	284	14	j	j	PROPN
cana-5209	284	15	(	(	PUNCT
cana-5209	284	16	x0	x0	PROPN
cana-5209	284	17	,	,	PUNCT
cana-5209	284	18	y0	y0	PROPN
cana-5209	284	19	)	)	PUNCT
cana-5209	284	20	≧	≧	SYM
cana-5209	284	21	c(x0,y0	c(x0,y0	NOUN
cana-5209	284	22	)	)	PUNCT
cana-5209	284	23	1	1	PROPN
cana-5209	284	24	τ	τ	X
cana-5209	284	25			PROPN
cana-5209	284	26	p∑	p∑	PROPN
cana-5209	284	27	i=1	i=1	PROPN
cana-5209	284	28	ui∇fi(y0	ui∇fi(y0	PROPN
cana-5209	284	29	)	)	PUNCT
cana-5209	285	1	+	+	CCONJ
cana-5209	285	2	∑	∑	PROPN
cana-5209	285	3	j∈j	j∈j	NOUN
cana-5209	285	4	vj∇gj(y0	vj∇gj(y0	NOUN
cana-5209	285	5	)	)	PUNCT
cana-5209	285	6			PROPN
cana-5209	285	7	η(x0	η(x0	NOUN
cana-5209	285	8	,	,	PUNCT
cana-5209	285	9	y0	y0	NOUN
cana-5209	285	10	)	)	PUNCT
cana-5209	285	11	+	+	CCONJ
cana-5209	285	12	1	1	NUM
cana-5209	285	13	τ	τ	X
cana-5209	286	1			PROPN
cana-5209	286	2	p∑	p∑	PRON
cana-5209	286	3	i=1	i=1	PRON
cana-5209	286	4	uiρ	uiρ	ADJ
cana-5209	286	5	1	1	NUM
cana-5209	287	1	i	i	PRON
cana-5209	287	2	d1i	d1i	VERB
cana-5209	287	3	(	(	PUNCT
cana-5209	287	4	x0	x0	PROPN
cana-5209	287	5	,	,	PUNCT
cana-5209	287	6	y0	y0	PROPN
cana-5209	287	7	)	)	PUNCT
cana-5209	287	8	α1	α1	PROPN
cana-5209	287	9	i	i	PRON
cana-5209	287	10	(	(	PUNCT
cana-5209	287	11	x0	x0	PROPN
cana-5209	287	12	,	,	PUNCT
cana-5209	287	13	y0	y0	PROPN
cana-5209	287	14	)	)	PUNCT
cana-5209	288	1	+	+	CCONJ
cana-5209	288	2	∑	∑	PROPN
cana-5209	288	3	j∈j	j∈j	ADJ
cana-5209	288	4	vjρ	vjρ	PROPN
cana-5209	288	5	2	2	NUM
cana-5209	288	6	j	j	PROPN
cana-5209	288	7	d2j	d2j	PROPN
cana-5209	288	8	(	(	PUNCT
cana-5209	288	9	x0	x0	PROPN
cana-5209	288	10	,	,	PUNCT
cana-5209	288	11	y0	y0	PROPN
cana-5209	288	12	)	)	PUNCT
cana-5209	288	13	α2	α2	PROPN
cana-5209	288	14	j	j	PROPN
cana-5209	288	15	(	(	PUNCT
cana-5209	288	16	x0	x0	PROPN
cana-5209	288	17	,	,	PUNCT
cana-5209	288	18	y0	y0	PROPN
cana-5209	288	19	)	)	PUNCT
cana-5209	288	20			PROPN
cana-5209	288	21	≧	≧	X
cana-5209	288	22	0	0	NUM
cana-5209	288	23	.	.	NOUN
cana-5209	288	24	which	which	PRON
cana-5209	288	25	gives	give	VERB
cana-5209	288	26	a	a	DET
cana-5209	288	27	contradiction	contradiction	NOUN
cana-5209	288	28	.	.	PUNCT
cana-5209	289	1	thus	thus	ADV
cana-5209	289	2	,	,	PUNCT
cana-5209	289	3	the	the	DET
cana-5209	289	4	proof	proof	NOUN
cana-5209	289	5	of	of	ADP
cana-5209	289	6	part	part	NOUN
cana-5209	289	7	(	(	PUNCT
cana-5209	289	8	a	a	NOUN
cana-5209	289	9	)	)	PUNCT
cana-5209	289	10	is	be	AUX
cana-5209	289	11	complete	complete	ADJ
cana-5209	289	12	.	.	PUNCT
cana-5209	290	1	similarly	similarly	ADV
cana-5209	290	2	we	we	PRON
cana-5209	290	3	can	can	AUX
cana-5209	290	4	prove	prove	VERB
cana-5209	290	5	part	part	NOUN
cana-5209	290	6	(	(	PUNCT
cana-5209	290	7	b	b	NOUN
cana-5209	290	8	)	)	PUNCT
cana-5209	290	9	(	(	PUNCT
cana-5209	290	10	e	e	NOUN
cana-5209	290	11	)	)	PUNCT
cana-5209	290	12	.	.	PUNCT
cana-5209	291	1	theorem	theorem	VERB
cana-5209	291	2	5.2	5.2	NUM
cana-5209	291	3	.	.	PUNCT
cana-5209	292	1	(	(	PUNCT
cana-5209	292	2	strong	strong	ADJ
cana-5209	292	3	duality	duality	NOUN
cana-5209	292	4	)	)	PUNCT
cana-5209	292	5	let	let	VERB
cana-5209	292	6	f	f	PROPN
cana-5209	292	7	and	and	CCONJ
cana-5209	292	8	vt	vt	PROPN
cana-5209	292	9	g	g	PROPN
cana-5209	292	10	be	be	AUX
cana-5209	292	11	satisfy	satisfy	NOUN
cana-5209	292	12	any	any	PRON
cana-5209	292	13	of	of	ADP
cana-5209	292	14	the	the	DET
cana-5209	292	15	five	five	NUM
cana-5209	292	16	assumptions	assumption	NOUN
cana-5209	292	17	specified	specify	VERB
cana-5209	292	18	in	in	ADP
cana-5209	292	19	theorem	theorem	NOUN
cana-5209	292	20	5.1	5.1	NUM
cana-5209	292	21	.	.	PUNCT
cana-5209	293	1	if	if	SCONJ
cana-5209	293	2	x0	x0	PROPN
cana-5209	293	3	∈	∈	PROPN
cana-5209	293	4	s	s	VERB
cana-5209	293	5	is	be	AUX
cana-5209	293	6	an	an	DET
cana-5209	293	7	efficient	efficient	ADJ
cana-5209	293	8	solution	solution	NOUN
cana-5209	293	9	for	for	ADP
cana-5209	293	10	(	(	PUNCT
cana-5209	293	11	sp	sp	NOUN
cana-5209	293	12	)	)	PUNCT
cana-5209	293	13	and	and	CCONJ
cana-5209	293	14	(	(	PUNCT
cana-5209	293	15	sp	sp	NOUN
cana-5209	293	16	)	)	PUNCT
cana-5209	293	17	satisfies	satisfy	VERB
cana-5209	293	18	a	a	DET
cana-5209	293	19	suitable	suitable	ADJ
cana-5209	293	20	constraint	constraint	NOUN
cana-5209	293	21	qualification	qualification	NOUN
cana-5209	293	22	(	(	PUNCT
cana-5209	293	23	see[26	see[26	ADP
cana-5209	293	24	]	]	PUNCT
cana-5209	293	25	)	)	PUNCT
cana-5209	293	26	.	.	PUNCT
cana-5209	294	1	then	then	ADV
cana-5209	294	2	,	,	PUNCT
cana-5209	294	3	∃	∃	PROPN
cana-5209	294	4	u	u	PROPN
cana-5209	294	5	∈	∈	PROPN
cana-5209	294	6	rp	rp	NOUN
cana-5209	294	7	+	+	CCONJ
cana-5209	294	8	u	u	X
cana-5209	294	9	>	>	X
cana-5209	294	10	0	0	NUM
cana-5209	294	11	,	,	PUNCT
cana-5209	294	12	v	v	NOUN
cana-5209	294	13	=	=	SYM
cana-5209	294	14	(	(	PUNCT
cana-5209	294	15	vj)(j∈j	vj)(j∈j	ADV
cana-5209	294	16	)	)	PUNCT
cana-5209	294	17	,	,	PUNCT
cana-5209	294	18	vj	vj	PROPN
cana-5209	294	19	∈	∈	PROPN
cana-5209	294	20	r+	r+	NOUN
cana-5209	294	21	,	,	PUNCT
cana-5209	294	22	such	such	ADJ
cana-5209	294	23	that	that	PRON
cana-5209	294	24	(	(	PUNCT
cana-5209	294	25	x0	x0	PROPN
cana-5209	294	26	,	,	PUNCT
cana-5209	294	27	u	u	NOUN
cana-5209	294	28	,	,	PUNCT
cana-5209	294	29	v	v	NOUN
cana-5209	294	30	)	)	PUNCT
cana-5209	294	31	is	be	AUX
cana-5209	294	32	an	an	DET
cana-5209	294	33	efficient	efficient	ADJ
cana-5209	294	34	solution	solution	NOUN
cana-5209	294	35	of	of	ADP
cana-5209	294	36	(	(	PUNCT
cana-5209	294	37	mwd	mwd	PROPN
cana-5209	294	38	)	)	PUNCT
cana-5209	294	39	and	and	CCONJ
cana-5209	294	40	the	the	DET
cana-5209	294	41	respective	respective	ADJ
cana-5209	294	42	objective	objective	ADJ
cana-5209	294	43	values	value	NOUN
cana-5209	294	44	of	of	ADP
cana-5209	294	45	(	(	PUNCT
cana-5209	294	46	sp	sp	NOUN
cana-5209	294	47	)	)	PUNCT
cana-5209	294	48	and	and	CCONJ
cana-5209	294	49	(	(	PUNCT
cana-5209	294	50	mwd	mwd	PROPN
cana-5209	294	51	)	)	PUNCT
cana-5209	294	52	are	be	AUX
cana-5209	294	53	equal	equal	ADJ
cana-5209	294	54	.	.	PUNCT
cana-5209	295	1	https://internationalpubls.com	https://internationalpubls.com	X
cana-5209	295	2	1148	1148	NUM
cana-5209	295	3	communications	communication	NOUN
cana-5209	295	4	on	on	ADP
cana-5209	295	5	applied	apply	VERB
cana-5209	295	6	nonlinear	nonlinear	ADJ
cana-5209	295	7	analysis	analysis	NOUN
cana-5209	295	8	issn	issn	NOUN
cana-5209	295	9	:	:	PUNCT
cana-5209	295	10	1074	1074	NUM
cana-5209	295	11	-	-	PUNCT
cana-5209	295	12	133x	133x	NUM
cana-5209	295	13	vol	vol	VERB
cana-5209	295	14	32	32	NUM
cana-5209	295	15	no	no	NOUN
cana-5209	295	16	.	.	PUNCT
cana-5209	296	1	10s	10	NOUN
cana-5209	296	2	(	(	PUNCT
cana-5209	296	3	2025	2025	NUM
cana-5209	296	4	)	)	PUNCT
cana-5209	296	5	proof	proof	NOUN
cana-5209	296	6	.	.	PUNCT
cana-5209	297	1	since	since	SCONJ
cana-5209	297	2	x0	x0	PROPN
cana-5209	297	3	∈	∈	PROPN
cana-5209	297	4	s	s	VERB
cana-5209	297	5	is	be	AUX
cana-5209	297	6	an	an	DET
cana-5209	297	7	efficient	efficient	ADJ
cana-5209	297	8	solution	solution	NOUN
cana-5209	297	9	for	for	ADP
cana-5209	297	10	(	(	PUNCT
cana-5209	297	11	sp	sp	NOUN
cana-5209	297	12	)	)	PUNCT
cana-5209	297	13	and	and	CCONJ
cana-5209	297	14	a	a	DET
cana-5209	297	15	suitable	suitable	ADJ
cana-5209	297	16	constraint	constraint	NOUN
cana-5209	297	17	qualification	qualification	NOUN
cana-5209	297	18	is	be	AUX
cana-5209	297	19	satisfied	satisfied	ADJ
cana-5209	297	20	(	(	PUNCT
cana-5209	297	21	see[26	see[26	ADP
cana-5209	297	22	]	]	PUNCT
cana-5209	297	23	)	)	PUNCT
cana-5209	297	24	.	.	PUNCT
cana-5209	298	1	therefore	therefore	ADV
cana-5209	298	2	by	by	ADP
cana-5209	298	3	theorem	theorem	NOUN
cana-5209	298	4	4.1	4.1	NUM
cana-5209	298	5	,	,	PUNCT
cana-5209	298	6	there	there	PRON
cana-5209	298	7	exist	exist	VERB
cana-5209	298	8	u	u	PROPN
cana-5209	298	9	∈	∈	NOUN
cana-5209	298	10	r+	r+	PUNCT
cana-5209	298	11	p	p	PROPN
cana-5209	298	12	u	u	X
cana-5209	298	13	>	>	X
cana-5209	298	14	0	0	NUM
cana-5209	298	15	,	,	PUNCT
cana-5209	298	16	v	v	NOUN
cana-5209	298	17	=	=	SYM
cana-5209	298	18	(	(	PUNCT
cana-5209	298	19	vj)(j∈j	vj)(j∈j	ADV
cana-5209	298	20	)	)	PUNCT
cana-5209	298	21	,	,	PUNCT
cana-5209	298	22	vj	vj	PROPN
cana-5209	298	23	∈	∈	PROPN
cana-5209	298	24	r+	r+	NOUN
cana-5209	298	25	,	,	PUNCT
cana-5209	298	26	such	such	ADJ
cana-5209	298	27	that	that	PRON
cana-5209	298	28	(	(	PUNCT
cana-5209	298	29	x0	x0	PROPN
cana-5209	298	30	,	,	PUNCT
cana-5209	298	31	u	u	NOUN
cana-5209	298	32	,	,	PUNCT
cana-5209	298	33	v	v	NOUN
cana-5209	298	34	)	)	PUNCT
cana-5209	298	35	is	be	AUX
cana-5209	298	36	a	a	DET
cana-5209	298	37	feasible	feasible	ADJ
cana-5209	298	38	solution	solution	NOUN
cana-5209	298	39	for	for	ADP
cana-5209	298	40	(	(	PUNCT
cana-5209	298	41	mwd	mwd	PROPN
cana-5209	298	42	)	)	PUNCT
cana-5209	298	43	.	.	PUNCT
cana-5209	299	1	suppose	suppose	VERB
cana-5209	299	2	that	that	SCONJ
cana-5209	299	3	(	(	PUNCT
cana-5209	299	4	x0	x0	PROPN
cana-5209	299	5	,	,	PUNCT
cana-5209	299	6	u	u	NOUN
cana-5209	299	7	,	,	PUNCT
cana-5209	299	8	v	v	NOUN
cana-5209	299	9	)	)	PUNCT
cana-5209	299	10	is	be	AUX
cana-5209	299	11	not	not	PART
cana-5209	299	12	an	an	DET
cana-5209	299	13	efficient	efficient	ADJ
cana-5209	299	14	solution	solution	NOUN
cana-5209	299	15	for	for	ADP
cana-5209	299	16	(	(	PUNCT
cana-5209	299	17	mwd	mwd	PROPN
cana-5209	299	18	)	)	PUNCT
cana-5209	299	19	,	,	PUNCT
cana-5209	299	20	this	this	PRON
cana-5209	299	21	implies	imply	VERB
cana-5209	299	22	there	there	PRON
cana-5209	299	23	exist	exist	VERB
cana-5209	299	24	a	a	DET
cana-5209	299	25	feasible	feasible	ADJ
cana-5209	299	26	solution	solution	NOUN
cana-5209	299	27	(	(	PUNCT
cana-5209	299	28	y0	y0	NOUN
cana-5209	299	29	,	,	PUNCT
cana-5209	299	30	u0	u0	ADJ
cana-5209	299	31	,	,	PUNCT
cana-5209	299	32	v0	v0	PROPN
cana-5209	299	33	)	)	PUNCT
cana-5209	299	34	for	for	ADP
cana-5209	299	35	(	(	PUNCT
cana-5209	299	36	mwd	mwd	PROPN
cana-5209	299	37	)	)	PUNCT
cana-5209	299	38	,	,	PUNCT
cana-5209	299	39	s.t	s.t	PROPN
cana-5209	299	40	.	.	PROPN
cana-5209	299	41	(	(	PUNCT
cana-5209	299	42	f1(x0	f1(x0	NOUN
cana-5209	299	43	)	)	PUNCT
cana-5209	299	44	,	,	PUNCT
cana-5209	299	45	...	...	PUNCT
cana-5209	299	46	,	,	PUNCT
cana-5209	299	47	fp(x0	fp(x0	NOUN
cana-5209	299	48	)	)	PUNCT
cana-5209	299	49	)	)	PUNCT
cana-5209	299	50	≤	≤	NOUN
cana-5209	299	51	(	(	PUNCT
cana-5209	299	52	f1(y0	f1(y0	NOUN
cana-5209	299	53	)	)	PUNCT
cana-5209	299	54	,	,	PUNCT
cana-5209	299	55	...	...	PUNCT
cana-5209	299	56	,	,	PUNCT
cana-5209	299	57	fp(y0	fp(y0	NOUN
cana-5209	299	58	)	)	PUNCT
cana-5209	299	59	)	)	PUNCT
cana-5209	299	60	,	,	PUNCT
cana-5209	299	61	which	which	PRON
cana-5209	299	62	contradicts	contradict	VERB
cana-5209	299	63	theorem	theorem	VERB
cana-5209	299	64	5.1	5.1	NUM
cana-5209	299	65	.	.	PUNCT
cana-5209	300	1	hence	hence	ADV
cana-5209	300	2	,	,	PUNCT
cana-5209	300	3	the	the	DET
cana-5209	300	4	proof	proof	NOUN
cana-5209	300	5	is	be	AUX
cana-5209	300	6	complete	complete	ADJ
cana-5209	300	7	.	.	PUNCT
cana-5209	301	1	theorem	theorem	VERB
cana-5209	301	2	5.3	5.3	NUM
cana-5209	301	3	.	.	PUNCT
cana-5209	302	1	(	(	PUNCT
cana-5209	302	2	strict	strict	ADJ
cana-5209	302	3	converse	converse	NOUN
cana-5209	302	4	duality	duality	NOUN
cana-5209	302	5	)	)	PUNCT
cana-5209	302	6	let	let	VERB
cana-5209	302	7	the	the	DET
cana-5209	302	8	assumptions	assumption	NOUN
cana-5209	302	9	of	of	ADP
cana-5209	302	10	theorem	theorem	ADJ
cana-5209	302	11	5.2	5.2	NUM
cana-5209	302	12	be	be	AUX
cana-5209	302	13	satisfied	satisfied	ADJ
cana-5209	302	14	and	and	CCONJ
cana-5209	302	15	f	f	PROPN
cana-5209	302	16	be	be	AUX
cana-5209	302	17	strictly	strictly	ADV
cana-5209	302	18	(	(	PUNCT
cana-5209	302	19	c	c	X
cana-5209	302	20	,	,	PUNCT
cana-5209	302	21	α1	α1	PROPN
cana-5209	302	22	,	,	PUNCT
cana-5209	302	23	η1	η1	NOUN
cana-5209	302	24	,	,	PUNCT
cana-5209	302	25	ρ1	ρ1	NOUN
cana-5209	302	26	,	,	PUNCT
cana-5209	302	27	d1)-pseudo	d1)-pseudo	NOUN
cana-5209	302	28	-	-	NOUN
cana-5209	302	29	invex	invex	NOUN
cana-5209	302	30	at	at	ADP
cana-5209	302	31	y0	y0	NOUN
cana-5209	302	32	.	.	PUNCT
cana-5209	303	1	if	if	SCONJ
cana-5209	303	2	x0	x0	PROPN
cana-5209	303	3	and	and	CCONJ
cana-5209	303	4	(	(	PUNCT
cana-5209	303	5	y0	y0	NOUN
cana-5209	303	6	,	,	PUNCT
cana-5209	303	7	u0	u0	ADJ
cana-5209	303	8	,	,	PUNCT
cana-5209	303	9	v0	v0	PROPN
cana-5209	303	10	)	)	PUNCT
cana-5209	303	11	are	be	AUX
cana-5209	303	12	feasible	feasible	ADJ
cana-5209	303	13	solutions	solution	NOUN
cana-5209	303	14	for	for	ADP
cana-5209	303	15	(	(	PUNCT
cana-5209	303	16	sp	sp	NOUN
cana-5209	303	17	)	)	PUNCT
cana-5209	303	18	and	and	CCONJ
cana-5209	303	19	(	(	PUNCT
cana-5209	303	20	mwd	mwd	PROPN
cana-5209	303	21	)	)	PUNCT
cana-5209	303	22	respectively	respectively	ADV
cana-5209	303	23	,	,	PUNCT
cana-5209	303	24	then	then	ADV
cana-5209	303	25	x0	x0	PROPN
cana-5209	303	26	=	=	SYM
cana-5209	303	27	y0	y0	NOUN
cana-5209	303	28	.	.	PUNCT
cana-5209	304	1	proof	proof	NOUN
cana-5209	304	2	.	.	PUNCT
cana-5209	305	1	suppose	suppose	VERB
cana-5209	305	2	that	that	SCONJ
cana-5209	305	3	x0	x0	PROPN
cana-5209	305	4	̸=	̸=	PROPN
cana-5209	305	5	y0	y0	NOUN
cana-5209	305	6	.	.	PUNCT
cana-5209	306	1	by	by	ADP
cana-5209	306	2	strong	strong	ADJ
cana-5209	306	3	duality	duality	NOUN
cana-5209	306	4	theorem	theorem	VERB
cana-5209	306	5	∃	∃	PROPN
cana-5209	306	6	u	u	NOUN
cana-5209	306	7	∈	∈	PROPN
cana-5209	306	8	rp	rp	NOUN
cana-5209	307	1	+	+	PROPN
cana-5209	307	2	,	,	PUNCT
cana-5209	307	3	u	u	NOUN
cana-5209	307	4	>	>	X
cana-5209	307	5	0	0	NUM
cana-5209	307	6	,	,	PUNCT
cana-5209	307	7	v	v	NOUN
cana-5209	307	8	=	=	SYM
cana-5209	307	9	(	(	PUNCT
cana-5209	307	10	vj)(j∈j	vj)(j∈j	ADV
cana-5209	307	11	)	)	PUNCT
cana-5209	307	12	,	,	PUNCT
cana-5209	307	13	vj	vj	PROPN
cana-5209	307	14	∈	∈	PROPN
cana-5209	307	15	r+	r+	NOUN
cana-5209	307	16	,	,	PUNCT
cana-5209	307	17	such	such	ADJ
cana-5209	307	18	that	that	PRON
cana-5209	307	19	(	(	PUNCT
cana-5209	307	20	x0	x0	PROPN
cana-5209	307	21	,	,	PUNCT
cana-5209	307	22	u	u	NOUN
cana-5209	307	23	,	,	PUNCT
cana-5209	307	24	v	v	NOUN
cana-5209	307	25	)	)	PUNCT
cana-5209	307	26	is	be	AUX
cana-5209	307	27	an	an	DET
cana-5209	307	28	efficient	efficient	ADJ
cana-5209	307	29	solution	solution	NOUN
cana-5209	307	30	for	for	ADP
cana-5209	307	31	(	(	PUNCT
cana-5209	307	32	mwd	mwd	PROPN
cana-5209	307	33	)	)	PUNCT
cana-5209	307	34	.	.	PUNCT
cana-5209	308	1	hence	hence	ADV
cana-5209	308	2	,	,	PUNCT
cana-5209	308	3	f(x0	f(x0	PROPN
cana-5209	308	4	)	)	PUNCT
cana-5209	308	5	=	=	SYM
cana-5209	308	6	f(y0	f(y0	NOUN
cana-5209	308	7	)	)	PUNCT
cana-5209	308	8	.	.	PUNCT
cana-5209	309	1	(	(	PUNCT
cana-5209	309	2	5.6	5.6	NUM
cana-5209	309	3	)	)	PUNCT
cana-5209	309	4	since	since	SCONJ
cana-5209	309	5	x0	x0	PROPN
cana-5209	309	6	and	and	CCONJ
cana-5209	309	7	(	(	PUNCT
cana-5209	309	8	y0	y0	NOUN
cana-5209	309	9	,	,	PUNCT
cana-5209	309	10	u0	u0	ADJ
cana-5209	309	11	,	,	PUNCT
cana-5209	309	12	v0	v0	PROPN
cana-5209	309	13	)	)	PUNCT
cana-5209	309	14	are	be	AUX
cana-5209	309	15	the	the	DET
cana-5209	309	16	feasible	feasible	ADJ
cana-5209	309	17	solution	solution	NOUN
cana-5209	309	18	for	for	ADP
cana-5209	309	19	(	(	PUNCT
cana-5209	309	20	sp	sp	NOUN
cana-5209	309	21	)	)	PUNCT
cana-5209	309	22	and	and	CCONJ
cana-5209	309	23	(	(	PUNCT
cana-5209	309	24	mwd	mwd	PROPN
cana-5209	309	25	)	)	PUNCT
cana-5209	309	26	,	,	PUNCT
cana-5209	309	27	this	this	PRON
cana-5209	309	28	implies	imply	VERB
cana-5209	309	29	that	that	SCONJ
cana-5209	309	30	v0jgj(x0	v0jgj(x0	ADJ
cana-5209	309	31	)	)	PUNCT
cana-5209	309	32	≦	≦	NUM
cana-5209	309	33	0	0	NUM
cana-5209	309	34	≦	≦	NUM
cana-5209	309	35	v0jgj(y0	v0jgj(y0	NOUN
cana-5209	309	36	)	)	PUNCT
cana-5209	309	37	.	.	PUNCT
cana-5209	310	1	(	(	PUNCT
cana-5209	310	2	5.7	5.7	NUM
cana-5209	310	3	)	)	PUNCT
cana-5209	310	4	by	by	ADP
cana-5209	310	5	definition	definition	NOUN
cana-5209	310	6	of	of	ADP
cana-5209	310	7	(	(	PUNCT
cana-5209	310	8	c	c	NOUN
cana-5209	310	9	,	,	PUNCT
cana-5209	310	10	α2	α2	ADJ
cana-5209	310	11	,	,	PUNCT
cana-5209	310	12	η2	η2	NOUN
cana-5209	310	13	,	,	PUNCT
cana-5209	310	14	ρ2	ρ2	NOUN
cana-5209	310	15	,	,	PUNCT
cana-5209	310	16	d2)-quasi	d2)-quasi	NOUN
cana-5209	310	17	-	-	NOUN
cana-5209	310	18	invexity	invexity	NOUN
cana-5209	310	19	,	,	PUNCT
cana-5209	310	20	we	we	PRON
cana-5209	310	21	have	have	VERB
cana-5209	310	22	c(x0,y0)(v0j∇gj(y0))η(x0	c(x0,y0)(v0j∇gj(y0))η(x0	NUM
cana-5209	310	23	,	,	PUNCT
cana-5209	310	24	y0	y0	NOUN
cana-5209	310	25	)	)	PUNCT
cana-5209	310	26	+	+	CCONJ
cana-5209	310	27	ρ2j	ρ2j	NUM
cana-5209	310	28	d2j	d2j	NOUN
cana-5209	310	29	(	(	PUNCT
cana-5209	310	30	x0	x0	PROPN
cana-5209	310	31	,	,	PUNCT
cana-5209	310	32	y0	y0	PROPN
cana-5209	310	33	)	)	PUNCT
cana-5209	310	34	α2	α2	PROPN
cana-5209	310	35	j	j	PROPN
cana-5209	310	36	(	(	PUNCT
cana-5209	310	37	x0	x0	PROPN
cana-5209	310	38	,	,	PUNCT
cana-5209	310	39	y0	y0	PROPN
cana-5209	310	40	)	)	PUNCT
cana-5209	310	41	≦	≦	NOUN
cana-5209	310	42	0	0	NUM
cana-5209	310	43	,	,	PUNCT
cana-5209	310	44	(	(	PUNCT
cana-5209	310	45	η2	η2	X
cana-5209	310	46	=	=	SYM
cana-5209	310	47	η	η	NOUN
cana-5209	310	48	)	)	PUNCT
cana-5209	310	49	.	.	PUNCT
cana-5209	311	1	(	(	PUNCT
cana-5209	311	2	5.8	5.8	NUM
cana-5209	311	3	)	)	PUNCT
cana-5209	311	4	again	again	ADV
cana-5209	311	5	,	,	PUNCT
cana-5209	311	6	by	by	ADP
cana-5209	311	7	the	the	DET
cana-5209	311	8	assumption	assumption	NOUN
cana-5209	311	9	on	on	ADP
cana-5209	311	10	fi(i	fi(i	X
cana-5209	311	11	=	=	SYM
cana-5209	311	12	1	1	NUM
cana-5209	311	13	,	,	PUNCT
cana-5209	311	14	2	2	NUM
cana-5209	311	15	,	,	PUNCT
cana-5209	311	16	...	...	PUNCT
cana-5209	311	17	,	,	PUNCT
cana-5209	311	18	p	p	X
cana-5209	311	19	)	)	PUNCT
cana-5209	311	20	,	,	PUNCT
cana-5209	311	21	we	we	PRON
cana-5209	311	22	have	have	VERB
cana-5209	311	23	c(x0,y0)(∇fi(y0))η(x0	c(x0,y0)(∇fi(y0))η(x0	PROPN
cana-5209	311	24	,	,	PUNCT
cana-5209	311	25	y0	y0	PROPN
cana-5209	311	26	)	)	PUNCT
cana-5209	311	27	+	+	SYM
cana-5209	311	28	ρ1i	ρ1i	PUNCT
cana-5209	311	29	d1i	d1i	NOUN
cana-5209	311	30	(	(	PUNCT
cana-5209	311	31	x0	x0	PROPN
cana-5209	311	32	,	,	PUNCT
cana-5209	311	33	y0	y0	PROPN
cana-5209	311	34	)	)	PUNCT
cana-5209	311	35	α1	α1	PROPN
cana-5209	311	36	i	i	PRON
cana-5209	311	37	(	(	PUNCT
cana-5209	311	38	x0	x0	PROPN
cana-5209	311	39	,	,	PUNCT
cana-5209	311	40	y0	y0	PROPN
cana-5209	311	41	)	)	PUNCT
cana-5209	311	42	<	<	X
cana-5209	311	43	0	0	NUM
cana-5209	311	44	,	,	PUNCT
cana-5209	311	45	(	(	PUNCT
cana-5209	311	46	η1	η1	NOUN
cana-5209	311	47	=	=	SYM
cana-5209	311	48	η	η	PROPN
cana-5209	311	49	)	)	PUNCT
cana-5209	311	50	.	.	PUNCT
cana-5209	312	1	(	(	PUNCT
cana-5209	312	2	5.9	5.9	NUM
cana-5209	312	3	)	)	PUNCT
cana-5209	312	4	let	let	VERB
cana-5209	312	5	us	we	PRON
cana-5209	312	6	denote	denote	VERB
cana-5209	312	7	τ	τ	X
cana-5209	312	8	=	=	PUNCT
cana-5209	312	9	p∑	p∑	X
cana-5209	312	10	i=1	i=1	PRON
cana-5209	312	11	u0i	u0i	PROPN
cana-5209	313	1	+	+	CCONJ
cana-5209	313	2	∑	∑	PROPN
cana-5209	313	3	j∈i	j∈i	PROPN
cana-5209	313	4	v0j	v0j	PROPN
cana-5209	313	5	.	.	PUNCT
cana-5209	314	1	therefore	therefore	ADV
cana-5209	314	2	from	from	ADP
cana-5209	314	3	equations	equation	NOUN
cana-5209	314	4	(	(	PUNCT
cana-5209	314	5	5.8)-(5.9	5.8)-(5.9	NUM
cana-5209	314	6	)	)	PUNCT
cana-5209	314	7	and	and	CCONJ
cana-5209	314	8	the	the	DET
cana-5209	314	9	convexity	convexity	NOUN
cana-5209	314	10	of	of	ADP
cana-5209	314	11	c(x0,y0	c(x0,y0	NOUN
cana-5209	314	12	)	)	PUNCT
cana-5209	314	13	(	(	PUNCT
cana-5209	314	14	·	·	PUNCT
cana-5209	314	15	)	)	PUNCT
cana-5209	314	16	,	,	PUNCT
cana-5209	314	17	we	we	PRON
cana-5209	314	18	get	get	VERB
cana-5209	314	19	0	0	PUNCT
cana-5209	314	20	>	>	X
cana-5209	314	21	p∑	p∑	X
cana-5209	315	1	i=1	i=1	PROPN
cana-5209	315	2	u0i	u0i	ADV
cana-5209	316	1	τ	τ	PROPN
cana-5209	316	2	c(x0,y0)(∇fi(y0))η(x0	c(x0,y0)(∇fi(y0))η(x0	PROPN
cana-5209	316	3	,	,	PUNCT
cana-5209	316	4	y0	y0	PROPN
cana-5209	316	5	)	)	PUNCT
cana-5209	317	1	+	+	CCONJ
cana-5209	317	2	∑	∑	PROPN
cana-5209	317	3	j∈j	j∈j	PROPN
cana-5209	317	4	v0j	v0j	PROPN
cana-5209	317	5	τ	τ	PROPN
cana-5209	317	6	c(x0,y0)(∇gj(y0))η(x0	c(x0,y0)(∇gj(y0))η(x0	NOUN
cana-5209	317	7	,	,	PUNCT
cana-5209	317	8	y0	y0	PROPN
cana-5209	317	9	)	)	PUNCT
cana-5209	318	1	+	+	NUM
cana-5209	318	2	p∑	p∑	NOUN
cana-5209	318	3	i=1	i=1	PRON
cana-5209	318	4	u0i	u0i	ADV
cana-5209	318	5	τ	τ	X
cana-5209	318	6	ρ1i	ρ1i	PUNCT
cana-5209	318	7	d1i	d1i	PROPN
cana-5209	318	8	(	(	PUNCT
cana-5209	318	9	x0	x0	PROPN
cana-5209	318	10	,	,	PUNCT
cana-5209	318	11	y0	y0	PROPN
cana-5209	318	12	)	)	PUNCT
cana-5209	318	13	α1	α1	PROPN
cana-5209	318	14	i	i	PRON
cana-5209	318	15	(	(	PUNCT
cana-5209	318	16	x0	x0	PROPN
cana-5209	318	17	,	,	PUNCT
cana-5209	318	18	y0	y0	PROPN
cana-5209	318	19	)	)	PUNCT
cana-5209	318	20	+	+	CCONJ
cana-5209	318	21	∑	∑	PROPN
cana-5209	318	22	j∈j	j∈j	PROPN
cana-5209	318	23	v0j	v0j	PROPN
cana-5209	318	24	τ	τ	PROPN
cana-5209	318	25	ρ2j	ρ2j	SYM
cana-5209	318	26	d2j	d2j	X
cana-5209	318	27	(	(	PUNCT
cana-5209	318	28	x0	x0	PROPN
cana-5209	318	29	,	,	PUNCT
cana-5209	318	30	y0	y0	PROPN
cana-5209	318	31	)	)	PUNCT
cana-5209	318	32	α2	α2	PROPN
cana-5209	318	33	j	j	PROPN
cana-5209	318	34	(	(	PUNCT
cana-5209	318	35	x0	x0	PROPN
cana-5209	318	36	,	,	PUNCT
cana-5209	318	37	y0	y0	PROPN
cana-5209	318	38	)	)	PUNCT
cana-5209	318	39	≧	≧	SYM
cana-5209	318	40	c(x0,y0	c(x0,y0	NOUN
cana-5209	318	41	)	)	PUNCT
cana-5209	318	42	1	1	PROPN
cana-5209	318	43	τ	τ	X
cana-5209	319	1			PROPN
cana-5209	319	2	p∑	p∑	X
cana-5209	319	3	i=1	i=1	PRON
cana-5209	319	4	u0i∇fi(y0	u0i∇fi(y0	ADV
cana-5209	319	5	)	)	PUNCT
cana-5209	320	1	+	+	CCONJ
cana-5209	320	2	∑	∑	PROPN
cana-5209	320	3	j∈j	j∈j	PROPN
cana-5209	320	4	v0j∇gj(y0	v0j∇gj(y0	PROPN
cana-5209	320	5	)	)	PUNCT
cana-5209	320	6			PROPN
cana-5209	320	7	η(x0	η(x0	NOUN
cana-5209	320	8	,	,	PUNCT
cana-5209	320	9	y0	y0	NOUN
cana-5209	320	10	)	)	PUNCT
cana-5209	320	11	+	+	CCONJ
cana-5209	320	12	1	1	NUM
cana-5209	320	13	τ	τ	X
cana-5209	321	1			X
cana-5209	321	2	p∑	p∑	PROPN
cana-5209	321	3	i=1	i=1	PROPN
cana-5209	321	4	u0iρ	u0iρ	PROPN
cana-5209	321	5	1	1	NUM
cana-5209	321	6	i	i	PRON
cana-5209	321	7	d1i	d1i	PROPN
cana-5209	321	8	(	(	PUNCT
cana-5209	321	9	x0	x0	PROPN
cana-5209	321	10	,	,	PUNCT
cana-5209	321	11	y0	y0	PROPN
cana-5209	321	12	)	)	PUNCT
cana-5209	321	13	α1	α1	PROPN
cana-5209	321	14	i	i	PRON
cana-5209	321	15	(	(	PUNCT
cana-5209	321	16	x0	x0	PROPN
cana-5209	321	17	,	,	PUNCT
cana-5209	321	18	y0	y0	PROPN
cana-5209	321	19	)	)	PUNCT
cana-5209	321	20	+	+	CCONJ
cana-5209	321	21	∑	∑	PROPN
cana-5209	321	22	j∈j	j∈j	NOUN
cana-5209	321	23	v0jρ	v0jρ	AUX
cana-5209	321	24	2	2	NUM
cana-5209	321	25	j	j	PROPN
cana-5209	321	26	d2j	d2j	PROPN
cana-5209	321	27	(	(	PUNCT
cana-5209	321	28	x0	x0	PROPN
cana-5209	321	29	,	,	PUNCT
cana-5209	321	30	y0	y0	PROPN
cana-5209	321	31	)	)	PUNCT
cana-5209	321	32	α2	α2	PROPN
cana-5209	321	33	j	j	PROPN
cana-5209	321	34	(	(	PUNCT
cana-5209	321	35	x0	x0	PROPN
cana-5209	321	36	,	,	PUNCT
cana-5209	321	37	y0	y0	PROPN
cana-5209	321	38	)	)	PUNCT
cana-5209	321	39			PROPN
cana-5209	321	40	≧	≧	X
cana-5209	321	41	0	0	NUM
cana-5209	321	42	.	.	NOUN
cana-5209	321	43	which	which	PRON
cana-5209	321	44	gives	give	VERB
cana-5209	321	45	a	a	DET
cana-5209	321	46	contradiction	contradiction	NOUN
cana-5209	321	47	,	,	PUNCT
cana-5209	321	48	i.e.	i.e.	X
cana-5209	321	49	,	,	PUNCT
cana-5209	321	50	our	our	PRON
cana-5209	321	51	assumption	assumption	NOUN
cana-5209	321	52	is	be	AUX
cana-5209	321	53	wrong	wrong	ADJ
cana-5209	321	54	.	.	PUNCT
cana-5209	322	1	therefore	therefore	ADV
cana-5209	322	2	x0	x0	PROPN
cana-5209	322	3	=	=	SYM
cana-5209	322	4	y0	y0	NOUN
cana-5209	322	5	.	.	PUNCT
cana-5209	323	1	6	6	NUM
cana-5209	323	2	conclusion	conclusion	NOUN
cana-5209	323	3	in	in	ADP
cana-5209	323	4	this	this	DET
cana-5209	323	5	paper	paper	NOUN
cana-5209	323	6	,	,	PUNCT
cana-5209	323	7	we	we	PRON
cana-5209	323	8	formulted	formulte	VERB
cana-5209	323	9	generalized	generalized	ADJ
cana-5209	323	10	(	(	PUNCT
cana-5209	323	11	c	c	X
cana-5209	323	12	,	,	PUNCT
cana-5209	323	13	α	α	PROPN
cana-5209	323	14	,	,	PUNCT
cana-5209	323	15	η	η	PROPN
cana-5209	323	16	,	,	PUNCT
cana-5209	323	17	ρ	ρ	PROPN
cana-5209	323	18	,	,	PUNCT
cana-5209	323	19	d	d	NOUN
cana-5209	323	20	)	)	PUNCT
cana-5209	323	21	invexities	invexitie	NOUN
cana-5209	323	22	,	,	PUNCT
cana-5209	323	23	which	which	PRON
cana-5209	323	24	are	be	AUX
cana-5209	323	25	the	the	DET
cana-5209	323	26	generalized	generalized	ADJ
cana-5209	323	27	version	version	NOUN
cana-5209	323	28	of	of	ADP
cana-5209	323	29	(	(	PUNCT
cana-5209	323	30	c	c	PROPN
cana-5209	323	31	,	,	PUNCT
cana-5209	323	32	α	α	PROPN
cana-5209	323	33	,	,	PUNCT
cana-5209	323	34	η	η	PROPN
cana-5209	323	35	,	,	PUNCT
cana-5209	323	36	ρ	ρ	PROPN
cana-5209	323	37	,	,	PUNCT
cana-5209	323	38	d	d	NOUN
cana-5209	323	39	)	)	PUNCT
cana-5209	323	40	convexity	convexity	NOUN
cana-5209	323	41	.	.	PUNCT
cana-5209	324	1	using	use	VERB
cana-5209	324	2	these	these	DET
cana-5209	324	3	definitions	definition	NOUN
cana-5209	324	4	,	,	PUNCT
cana-5209	324	5	we	we	PRON
cana-5209	324	6	have	have	AUX
cana-5209	324	7	stablished	stablishe	VERB
cana-5209	324	8	several	several	ADJ
cana-5209	324	9	optimality	optimality	NOUN
cana-5209	324	10	and	and	CCONJ
cana-5209	324	11	duality	duality	NOUN
cana-5209	324	12	results	result	NOUN
cana-5209	324	13	for	for	ADP
cana-5209	324	14	multiobjective	multiobjective	ADJ
cana-5209	324	15	semiinfinite	semiinfinite	ADJ
cana-5209	324	16	programming	programming	NOUN
cana-5209	324	17	problems	problem	NOUN
cana-5209	324	18	.	.	PUNCT
cana-5209	325	1	the	the	DET
cana-5209	325	2	results	result	NOUN
cana-5209	325	3	of	of	ADP
cana-5209	325	4	this	this	DET
cana-5209	325	5	paper	paper	NOUN
cana-5209	325	6	are	be	AUX
cana-5209	325	7	more	more	ADV
cana-5209	325	8	general	general	ADJ
cana-5209	325	9	than	than	ADP
cana-5209	325	10	the	the	DET
cana-5209	325	11	corresponding	corresponding	ADJ
cana-5209	325	12	results	result	NOUN
cana-5209	325	13	present	present	ADJ
cana-5209	325	14	in	in	ADP
cana-5209	325	15	the	the	DET
cana-5209	325	16	literature	literature	NOUN
cana-5209	325	17	[	[	X
cana-5209	325	18	19	19	NUM
cana-5209	325	19	]	]	PUNCT
cana-5209	325	20	.	.	PUNCT
cana-5209	326	1	these	these	DET
cana-5209	326	2	definitions	definition	NOUN
cana-5209	326	3	we	we	PRON
cana-5209	326	4	can	can	AUX
cana-5209	326	5	use	use	VERB
cana-5209	326	6	to	to	PART
cana-5209	326	7	extend	extend	VERB
cana-5209	326	8	more	more	ADJ
cana-5209	326	9	results	result	NOUN
cana-5209	326	10	of	of	ADP
cana-5209	326	11	literature	literature	NOUN
cana-5209	326	12	.	.	PUNCT
cana-5209	327	1	https://internationalpubls.com	https://internationalpubls.com	X
cana-5209	327	2	1149	1149	NUM
cana-5209	327	3	communications	communication	NOUN
cana-5209	327	4	on	on	ADP
cana-5209	327	5	applied	apply	VERB
cana-5209	327	6	nonlinear	nonlinear	ADJ
cana-5209	327	7	analysis	analysis	NOUN
cana-5209	327	8	issn	issn	NOUN
cana-5209	327	9	:	:	PUNCT
cana-5209	327	10	1074	1074	NUM
cana-5209	327	11	-	-	PUNCT
cana-5209	327	12	133x	133x	NUM
cana-5209	327	13	vol	vol	VERB
cana-5209	327	14	32	32	NUM
cana-5209	327	15	no	no	NOUN
cana-5209	327	16	.	.	PUNCT
cana-5209	328	1	10s	10	NOUN
cana-5209	328	2	(	(	PUNCT
cana-5209	328	3	2025	2025	NUM
cana-5209	328	4	)	)	PUNCT
cana-5209	328	5	acknowledgements	acknowledgement	NOUN
cana-5209	328	6	the	the	DET
cana-5209	328	7	authors	author	NOUN
cana-5209	328	8	wish	wish	VERB
cana-5209	328	9	to	to	PART
cana-5209	328	10	express	express	VERB
cana-5209	328	11	their	their	PRON
cana-5209	328	12	sincere	sincere	ADJ
cana-5209	328	13	gratitude	gratitude	NOUN
cana-5209	328	14	to	to	PART
cana-5209	328	15	prof	prof	PROPN
cana-5209	328	16	.	.	PUNCT
cana-5209	329	1	s.	s.	PROPN
cana-5209	329	2	k.	k.	PROPN
cana-5209	329	3	mishra	mishra	PROPN
cana-5209	329	4	from	from	ADP
cana-5209	329	5	department	department	PROPN
cana-5209	329	6	of	of	ADP
cana-5209	329	7	mathematics	mathematics	PROPN
cana-5209	329	8	,	,	PUNCT
cana-5209	329	9	banaras	banaras	PROPN
cana-5209	329	10	hindu	hindu	PROPN
cana-5209	329	11	university	university	PROPN
cana-5209	329	12	,	,	PUNCT
cana-5209	329	13	varanasi	varanasi	NOUN
cana-5209	329	14	,	,	PUNCT
cana-5209	329	15	for	for	ADP
cana-5209	329	16	his	his	PRON
cana-5209	329	17	valuable	valuable	ADJ
cana-5209	329	18	suggestions	suggestion	NOUN
cana-5209	329	19	that	that	PRON
cana-5209	329	20	have	have	AUX
cana-5209	329	21	significantly	significantly	ADV
cana-5209	329	22	enhanced	enhance	VERB
cana-5209	329	23	the	the	DET
cana-5209	329	24	research	research	NOUN
cana-5209	329	25	outcomes	outcome	NOUN
cana-5209	329	26	of	of	ADP
cana-5209	329	27	this	this	DET
cana-5209	329	28	article	article	NOUN
cana-5209	329	29	.	.	PUNCT
cana-5209	330	1	references	reference	NOUN
cana-5209	330	2	[	[	X
cana-5209	330	3	1	1	NUM
cana-5209	330	4	]	]	PUNCT
cana-5209	330	5	a.	a.	PROPN
cana-5209	330	6	ben	ben	PROPN
cana-5209	330	7	-	-	PROPN
cana-5209	330	8	israel	israel	PROPN
cana-5209	330	9	and	and	CCONJ
cana-5209	330	10	b.	b.	PROPN
cana-5209	330	11	mond	mond	PROPN
cana-5209	330	12	,	,	PUNCT
cana-5209	330	13	what	what	PRON
cana-5209	330	14	is	be	AUX
cana-5209	330	15	invexity	invexity	NOUN
cana-5209	330	16	,	,	PUNCT
cana-5209	330	17	journal	journal	NOUN
cana-5209	330	18	of	of	ADP
cana-5209	330	19	australian	australian	ADJ
cana-5209	330	20	mathematical	mathematical	ADJ
cana-5209	330	21	society	society	NOUN
cana-5209	330	22	,	,	PUNCT
cana-5209	330	23	28	28	NUM
cana-5209	330	24	b	b	PROPN
cana-5209	330	25	(	(	PUNCT
cana-5209	330	26	1986	1986	NUM
cana-5209	330	27	)	)	PUNCT
cana-5209	330	28	1	1	NUM
cana-5209	330	29	-	-	SYM
cana-5209	330	30	9	9	NUM
cana-5209	330	31	.	.	PUNCT
cana-5209	331	1	[	[	X
cana-5209	331	2	2	2	NUM
cana-5209	331	3	]	]	PUNCT
cana-5209	331	4	a.	a.	NOUN
cana-5209	331	5	charnes	charnes	PROPN
cana-5209	331	6	,	,	PUNCT
cana-5209	331	7	w.w	w.w	PROPN
cana-5209	331	8	.	.	PROPN
cana-5209	331	9	cooper	cooper	PROPN
cana-5209	331	10	,	,	PUNCT
cana-5209	331	11	k.o	k.o	PROPN
cana-5209	331	12	.	.	PROPN
cana-5209	331	13	kortanek	kortanek	PROPN
cana-5209	331	14	,	,	PUNCT
cana-5209	331	15	duality	duality	NOUN
cana-5209	331	16	,	,	PUNCT
cana-5209	331	17	haar	haar	PROPN
cana-5209	331	18	programs	program	NOUN
cana-5209	331	19	and	and	CCONJ
cana-5209	331	20	finite	finite	ADJ
cana-5209	331	21	sequence	sequence	NOUN
cana-5209	331	22	spaces	space	NOUN
cana-5209	331	23	.	.	PUNCT
cana-5209	332	1	proceedings	proceeding	NOUN
cana-5209	332	2	of	of	ADP
cana-5209	332	3	the	the	DET
cana-5209	332	4	national	national	PROPN
cana-5209	332	5	academy	academy	PROPN
cana-5209	332	6	of	of	ADP
cana-5209	332	7	science	science	PROPN
cana-5209	332	8	,	,	PUNCT
cana-5209	332	9	48	48	NUM
cana-5209	332	10	(	(	PUNCT
cana-5209	332	11	1962	1962	NUM
cana-5209	332	12	)	)	PUNCT
cana-5209	332	13	783	783	NUM
cana-5209	332	14	-	-	SYM
cana-5209	332	15	786	786	NUM
cana-5209	332	16	.	.	PUNCT
cana-5209	333	1	[	[	X
cana-5209	333	2	3	3	NUM
cana-5209	333	3	]	]	PUNCT
cana-5209	333	4	a.	a.	NOUN
cana-5209	333	5	charnes	charnes	PROPN
cana-5209	333	6	,	,	PUNCT
cana-5209	333	7	w.w	w.w	PROPN
cana-5209	333	8	.	.	PROPN
cana-5209	333	9	cooper	cooper	PROPN
cana-5209	333	10	,	,	PUNCT
cana-5209	333	11	k.o	k.o	PROPN
cana-5209	333	12	.	.	PROPN
cana-5209	333	13	kortanek	kortanek	PROPN
cana-5209	333	14	,	,	PUNCT
cana-5209	333	15	duality	duality	NOUN
cana-5209	333	16	in	in	ADP
cana-5209	333	17	semi	semi	ADJ
cana-5209	333	18	-	-	ADJ
cana-5209	333	19	infinite	infinite	ADJ
cana-5209	333	20	programs	program	NOUN
cana-5209	333	21	and	and	CCONJ
cana-5209	333	22	some	some	DET
cana-5209	333	23	works	work	NOUN
cana-5209	333	24	of	of	ADP
cana-5209	333	25	haar	haar	PROPN
cana-5209	333	26	and	and	CCONJ
cana-5209	333	27	carath´eodory	carath´eodory	NOUN
cana-5209	333	28	.	.	PUNCT
cana-5209	334	1	man	man	NOUN
cana-5209	334	2	.	.	PUNCT
cana-5209	335	1	sci	sci	PROPN
cana-5209	335	2	.	.	PROPN
cana-5209	335	3	,	,	PUNCT
cana-5209	335	4	9	9	NUM
cana-5209	335	5	(	(	PUNCT
cana-5209	335	6	1963	1963	NUM
cana-5209	335	7	)	)	PUNCT
cana-5209	335	8	209	209	NUM
cana-5209	335	9	-	-	SYM
cana-5209	335	10	228	228	NUM
cana-5209	335	11	.	.	PUNCT
cana-5209	336	1	[	[	X
cana-5209	336	2	4	4	NUM
cana-5209	336	3	]	]	PUNCT
cana-5209	336	4	a.	a.	NOUN
cana-5209	336	5	charnes	charnes	PROPN
cana-5209	336	6	,	,	PUNCT
cana-5209	336	7	w.w	w.w	PROPN
cana-5209	336	8	.	.	PROPN
cana-5209	336	9	cooper	cooper	PROPN
cana-5209	336	10	,	,	PUNCT
cana-5209	336	11	k.o	k.o	PROPN
cana-5209	336	12	.	.	PROPN
cana-5209	336	13	kortanek	kortanek	PROPN
cana-5209	336	14	,	,	PUNCT
cana-5209	336	15	on	on	ADP
cana-5209	336	16	the	the	DET
cana-5209	336	17	theory	theory	NOUN
cana-5209	336	18	of	of	ADP
cana-5209	336	19	semi	semi	ADJ
cana-5209	336	20	-	-	ADJ
cana-5209	336	21	infinite	infinite	ADJ
cana-5209	336	22	programming	programming	NOUN
cana-5209	336	23	and	and	CCONJ
cana-5209	336	24	a	a	DET
cana-5209	336	25	generalization	generalization	NOUN
cana-5209	336	26	of	of	ADP
cana-5209	336	27	the	the	DET
cana-5209	336	28	kuhn	kuhn	PROPN
cana-5209	336	29	-	-	PUNCT
cana-5209	336	30	tucker	tucker	PROPN
cana-5209	336	31	saddle	saddle	NOUN
cana-5209	336	32	point	point	NOUN
cana-5209	336	33	theorem	theorem	VERB
cana-5209	336	34	for	for	ADP
cana-5209	336	35	arbitrary	arbitrary	ADJ
cana-5209	336	36	convex	convex	NOUN
cana-5209	336	37	functions	function	NOUN
cana-5209	336	38	.	.	PUNCT
cana-5209	337	1	naval	naval	ADJ
cana-5209	337	2	research	research	PROPN
cana-5209	337	3	logistics	logistic	NOUN
cana-5209	337	4	quarterly	quarterly	ADV
cana-5209	337	5	,	,	PUNCT
cana-5209	337	6	16	16	NUM
cana-5209	337	7	(	(	PUNCT
cana-5209	337	8	1969	1969	NUM
cana-5209	337	9	)	)	PUNCT
cana-5209	337	10	41	41	NUM
cana-5209	337	11	-	-	SYM
cana-5209	337	12	51	51	NUM
cana-5209	337	13	.	.	PUNCT
cana-5209	338	1	[	[	X
cana-5209	338	2	5	5	NUM
cana-5209	338	3	]	]	PUNCT
cana-5209	338	4	a.	a.	PROPN
cana-5209	338	5	ismael	ismael	PROPN
cana-5209	338	6	f.	f.	PROPN
cana-5209	338	7	vaz	vaz	PROPN
cana-5209	338	8	,	,	PUNCT
cana-5209	338	9	edite	edite	PROPN
cana-5209	338	10	m.g.p	m.g.p	PROPN
cana-5209	338	11	.	.	PROPN
cana-5209	338	12	fernandes	fernandes	PROPN
cana-5209	338	13	,	,	PUNCT
cana-5209	338	14	m.	m.	NOUN
cana-5209	338	15	paula	paula	PROPN
cana-5209	338	16	s.f	s.f	PROPN
cana-5209	338	17	.	.	PROPN
cana-5209	338	18	gomes	gomes	PROPN
cana-5209	338	19	,	,	PUNCT
cana-5209	338	20	robot	robot	NOUN
cana-5209	338	21	trajectory	trajectory	NOUN
cana-5209	338	22	planning	planning	NOUN
cana-5209	338	23	with	with	ADP
cana-5209	338	24	semi	semi	ADJ
cana-5209	338	25	-	-	ADJ
cana-5209	338	26	infinite	infinite	ADJ
cana-5209	338	27	programming	programming	NOUN
cana-5209	338	28	.	.	PUNCT
cana-5209	339	1	european	european	PROPN
cana-5209	339	2	j.	j.	PROPN
cana-5209	339	3	oper	oper	PROPN
cana-5209	339	4	.	.	PUNCT
cana-5209	340	1	res	re	NOUN
cana-5209	340	2	.	.	PROPN
cana-5209	341	1	153	153	NUM
cana-5209	341	2	(	(	PUNCT
cana-5209	341	3	2004	2004	NUM
cana-5209	341	4	)	)	PUNCT
cana-5209	341	5	607	607	NUM
cana-5209	341	6	-	-	SYM
cana-5209	341	7	617	617	NUM
cana-5209	341	8	.	.	PUNCT
cana-5209	342	1	[	[	X
cana-5209	342	2	6	6	NUM
cana-5209	342	3	]	]	PUNCT
cana-5209	342	4	a.	a.	PROPN
cana-5209	342	5	ismael	ismael	PROPN
cana-5209	342	6	f.	f.	PROPN
cana-5209	342	7	vaz	vaz	PROPN
cana-5209	342	8	,	,	PUNCT
cana-5209	342	9	eug´enio	eug´enio	PROPN
cana-5209	342	10	c.	c.	PROPN
cana-5209	342	11	ferreira	ferreira	PROPN
cana-5209	342	12	,	,	PUNCT
cana-5209	342	13	air	air	NOUN
cana-5209	342	14	pollution	pollution	NOUN
cana-5209	342	15	control	control	NOUN
cana-5209	342	16	with	with	ADP
cana-5209	342	17	semi	semi	ADJ
cana-5209	342	18	-	-	ADJ
cana-5209	342	19	infinite	infinite	ADJ
cana-5209	342	20	programming	programming	NOUN
cana-5209	342	21	.	.	PUNCT
cana-5209	343	1	appl	appl	PROPN
cana-5209	343	2	.	.	PROPN
cana-5209	343	3	math	math	PROPN
cana-5209	343	4	.	.	PUNCT
cana-5209	344	1	model	model	PROPN
cana-5209	344	2	.	.	PROPN
cana-5209	345	1	,	,	PUNCT
cana-5209	345	2	33	33	NUM
cana-5209	345	3	(	(	PUNCT
cana-5209	345	4	2009	2009	NUM
cana-5209	345	5	)	)	PUNCT
cana-5209	345	6	1957	1957	NUM
cana-5209	345	7	-	-	SYM
cana-5209	345	8	1969	1969	NUM
cana-5209	345	9	.	.	PUNCT
cana-5209	346	1	[	[	X
cana-5209	346	2	7	7	X
cana-5209	346	3	]	]	PUNCT
cana-5209	346	4	a.	a.	NOUN
cana-5209	346	5	shapiro	shapiro	PROPN
cana-5209	346	6	,	,	PUNCT
cana-5209	346	7	semi	semi	ADJ
cana-5209	346	8	-	-	ADJ
cana-5209	346	9	infinite	infinite	ADJ
cana-5209	346	10	programming	programming	NOUN
cana-5209	346	11	,	,	PUNCT
cana-5209	346	12	duality	duality	NOUN
cana-5209	346	13	,	,	PUNCT
cana-5209	346	14	discretization	discretization	NOUN
cana-5209	346	15	and	and	CCONJ
cana-5209	346	16	optimality	optimality	NOUN
cana-5209	346	17	conditions	condition	NOUN
cana-5209	346	18	,	,	PUNCT
cana-5209	346	19	optimization	optimization	NOUN
cana-5209	346	20	:	:	PUNCT
cana-5209	346	21	a	a	DET
cana-5209	346	22	journal	journal	NOUN
cana-5209	346	23	of	of	ADP
cana-5209	346	24	mathematical	mathematical	ADJ
cana-5209	346	25	programming	programming	NOUN
cana-5209	346	26	and	and	CCONJ
cana-5209	346	27	operations	operation	NOUN
cana-5209	346	28	research	research	NOUN
cana-5209	346	29	,	,	PUNCT
cana-5209	346	30	58	58	NUM
cana-5209	346	31	(	(	PUNCT
cana-5209	346	32	2009	2009	NUM
cana-5209	346	33	)	)	PUNCT
cana-5209	346	34	133161	133161	NUM
cana-5209	346	35	.	.	PUNCT
cana-5209	347	1	[	[	X
cana-5209	347	2	8	8	NUM
cana-5209	347	3	]	]	PUNCT
cana-5209	347	4	a.	a.	NOUN
cana-5209	347	5	winterfeld	winterfeld	PROPN
cana-5209	347	6	,	,	PUNCT
cana-5209	347	7	application	application	NOUN
cana-5209	347	8	of	of	ADP
cana-5209	347	9	general	general	ADJ
cana-5209	347	10	semi	semi	ADJ
cana-5209	347	11	-	-	ADJ
cana-5209	347	12	infinite	infinite	ADJ
cana-5209	347	13	programming	programming	NOUN
cana-5209	347	14	to	to	ADP
cana-5209	347	15	lapidary	lapidary	ADJ
cana-5209	347	16	cutting	cutting	NOUN
cana-5209	347	17	problems	problem	NOUN
cana-5209	347	18	.	.	PUNCT
cana-5209	348	1	european	european	PROPN
cana-5209	348	2	j.	j.	PROPN
cana-5209	348	3	oper	oper	PROPN
cana-5209	348	4	.	.	PUNCT
cana-5209	348	5	res	res	PROPN
cana-5209	348	6	.	.	PROPN
cana-5209	348	7	,	,	PUNCT
cana-5209	348	8	191	191	NUM
cana-5209	348	9	(	(	PUNCT
cana-5209	348	10	2008	2008	NUM
cana-5209	348	11	)	)	PUNCT
cana-5209	348	12	838	838	NUM
cana-5209	348	13	-	-	SYM
cana-5209	348	14	854	854	NUM
cana-5209	348	15	.	.	PUNCT
cana-5209	349	1	[	[	X
cana-5209	349	2	9	9	NUM
cana-5209	349	3	]	]	X
cana-5209	349	4	b.d	b.d	PROPN
cana-5209	349	5	.	.	PROPN
cana-5209	349	6	craven	craven	PROPN
cana-5209	349	7	,	,	PUNCT
cana-5209	349	8	invex	invex	NOUN
cana-5209	349	9	functions	function	NOUN
cana-5209	349	10	and	and	CCONJ
cana-5209	349	11	constrained	constrain	VERB
cana-5209	349	12	local	local	ADJ
cana-5209	349	13	minima	minima	NOUN
cana-5209	349	14	,	,	PUNCT
cana-5209	349	15	bull	bull	NOUN
cana-5209	349	16	.	.	PUNCT
cana-5209	350	1	austral	austral	PROPN
cana-5209	350	2	.	.	PUNCT
cana-5209	351	1	math	math	NOUN
cana-5209	351	2	.	.	PUNCT
cana-5209	352	1	soc	soc	PROPN
cana-5209	352	2	.	.	PROPN
cana-5209	352	3	,	,	PUNCT
cana-5209	352	4	24	24	NUM
cana-5209	352	5	(	(	PUNCT
cana-5209	352	6	1981	1981	NUM
cana-5209	352	7	)	)	PUNCT
cana-5209	352	8	357	357	NUM
cana-5209	352	9	-	-	SYM
cana-5209	352	10	366	366	NUM
cana-5209	352	11	.	.	PUNCT
cana-5209	353	1	[	[	X
cana-5209	353	2	10	10	NUM
cana-5209	353	3	]	]	X
cana-5209	353	4	b.d	b.d	PROPN
cana-5209	353	5	.	.	PROPN
cana-5209	353	6	craven	craven	NOUN
cana-5209	353	7	and	and	CCONJ
cana-5209	353	8	b.m	b.m	PROPN
cana-5209	353	9	.	.	PROPN
cana-5209	353	10	glover	glover	PROPN
cana-5209	353	11	,	,	PUNCT
cana-5209	353	12	invex	invex	NOUN
cana-5209	353	13	functions	function	NOUN
cana-5209	353	14	and	and	CCONJ
cana-5209	353	15	duality	duality	NOUN
cana-5209	353	16	,	,	PUNCT
cana-5209	353	17	j.	j.	PROPN
cana-5209	353	18	austral	austral	PROPN
cana-5209	353	19	.	.	PUNCT
cana-5209	354	1	math	math	NOUN
cana-5209	354	2	.	.	PUNCT
cana-5209	355	1	soc	soc	PROPN
cana-5209	355	2	.	.	PROPN
cana-5209	355	3	,	,	PUNCT
cana-5209	355	4	24	24	NUM
cana-5209	355	5	(	(	PUNCT
cana-5209	355	6	1985	1985	NUM
cana-5209	355	7	)	)	PUNCT
cana-5209	355	8	1	1	NUM
cana-5209	355	9	-	-	SYM
cana-5209	355	10	20	20	NUM
cana-5209	355	11	.	.	PUNCT
cana-5209	356	1	[	[	X
cana-5209	356	2	11	11	NUM
cana-5209	356	3	]	]	X
cana-5209	356	4	belmiro	belmiro	PROPN
cana-5209	356	5	p.m.	p.m.	PROPN
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cana-5209	356	7	,	,	PUNCT
cana-5209	356	8	weng	weng	PROPN
cana-5209	356	9	kee	kee	PROPN
cana-5209	356	10	wong	wong	PROPN
cana-5209	356	11	,	,	PUNCT
cana-5209	356	12	a	a	DET
cana-5209	356	13	semi	semi	ADJ
cana-5209	356	14	-	-	ADJ
cana-5209	356	15	infinite	infinite	ADJ
cana-5209	356	16	programming	programming	NOUN
cana-5209	356	17	based	base	VERB
cana-5209	356	18	algorithm	algorithm	NOUN
cana-5209	356	19	for	for	ADP
cana-5209	356	20	finding	find	VERB
cana-5209	356	21	minimax	minimax	NOUN
cana-5209	356	22	optimal	optimal	ADJ
cana-5209	356	23	designs	design	NOUN
cana-5209	356	24	for	for	ADP
cana-5209	356	25	nonlinear	nonlinear	ADJ
cana-5209	356	26	models	model	NOUN
cana-5209	356	27	.	.	PUNCT
cana-5209	357	1	stat	stat	NOUN
cana-5209	357	2	.	.	PUNCT
cana-5209	358	1	comput	comput	NOUN
cana-5209	358	2	.	.	PUNCT
cana-5209	359	1	24	24	NUM
cana-5209	359	2	(	(	PUNCT
cana-5209	359	3	2014	2014	NUM
cana-5209	359	4	)	)	PUNCT
cana-5209	359	5	1063	1063	NUM
cana-5209	359	6	-	-	SYM
cana-5209	359	7	1080	1080	NUM
cana-5209	359	8	.	.	PUNCT
cana-5209	360	1	[	[	X
cana-5209	360	2	12	12	NUM
cana-5209	360	3	]	]	X
cana-5209	360	4	c.	c.	PROPN
cana-5209	360	5	li	li	PROPN
cana-5209	360	6	,	,	PUNCT
cana-5209	360	7	x.p	x.p	PROPN
cana-5209	360	8	.	.	PROPN
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cana-5209	360	10	,	,	PUNCT
cana-5209	360	11	y.h	y.h	PROPN
cana-5209	360	12	.	.	PROPN
cana-5209	360	13	hu	hu	PROPN
cana-5209	360	14	,	,	PUNCT
cana-5209	360	15	quasi	quasi	ADJ
cana-5209	360	16	-	-	NOUN
cana-5209	360	17	slater	slater	NOUN
cana-5209	360	18	and	and	CCONJ
cana-5209	360	19	farkas	farkas	ADJ
cana-5209	360	20	-	-	PUNCT
cana-5209	360	21	minkowski	minkowski	ADJ
cana-5209	360	22	qualifications	qualification	NOUN
cana-5209	360	23	for	for	ADP
cana-5209	360	24	semi	semi	ADJ
cana-5209	360	25	-	-	ADJ
cana-5209	360	26	infinite	infinite	ADJ
cana-5209	360	27	programming	programming	NOUN
cana-5209	360	28	with	with	ADP
cana-5209	360	29	applications	application	NOUN
cana-5209	360	30	.	.	PUNCT
cana-5209	361	1	siam	siam	PROPN
cana-5209	361	2	j.	j.	PROPN
cana-5209	361	3	optim	optim	PROPN
cana-5209	361	4	.	.	PUNCT
cana-5209	362	1	23	23	NUM
cana-5209	362	2	(	(	PUNCT
cana-5209	362	3	2013	2013	NUM
cana-5209	362	4	)	)	PUNCT
cana-5209	362	5	2208	2208	NUM
cana-5209	362	6	-	-	SYM
cana-5209	362	7	2230	2230	NUM
cana-5209	362	8	.	.	PUNCT
cana-5209	363	1	[	[	X
cana-5209	363	2	13	13	NUM
cana-5209	363	3	]	]	X
cana-5209	363	4	d.h	d.h	PROPN
cana-5209	363	5	.	.	PROPN
cana-5209	363	6	yuan	yuan	PROPN
cana-5209	363	7	,	,	PUNCT
cana-5209	363	8	x.l	x.l	PROPN
cana-5209	363	9	.	.	PUNCT
cana-5209	363	10	liu	liu	PROPN
cana-5209	363	11	,	,	PUNCT
cana-5209	363	12	a.	a.	NOUN
cana-5209	363	13	chinchuluun	chinchuluun	PROPN
cana-5209	363	14	and	and	CCONJ
cana-5209	363	15	p.m.	p.m.	NOUN
cana-5209	363	16	pardalos	pardalo	NOUN
cana-5209	363	17	,	,	PUNCT
cana-5209	363	18	nondifferentiable	nondifferentiable	ADJ
cana-5209	363	19	minimax	minimax	NOUN
cana-5209	363	20	fractional	fractional	ADJ
cana-5209	363	21	programming	programming	NOUN
cana-5209	363	22	problems	problem	NOUN
cana-5209	363	23	with	with	ADP
cana-5209	363	24	(	(	PUNCT
cana-5209	363	25	c	c	X
cana-5209	363	26	,	,	PUNCT
cana-5209	363	27	α	α	PROPN
cana-5209	363	28	,	,	PUNCT
cana-5209	363	29	ρ	ρ	NOUN
cana-5209	363	30	,	,	PUNCT
cana-5209	363	31	d)-convexity	d)-convexity	NOUN
cana-5209	363	32	,	,	PUNCT
cana-5209	363	33	j.	j.	PROPN
cana-5209	363	34	optim	optim	PROPN
cana-5209	363	35	.	.	PUNCT
cana-5209	364	1	theory	theory	NOUN
cana-5209	364	2	appl	appl	PROPN
cana-5209	364	3	.	.	PROPN
cana-5209	365	1	,	,	PUNCT
cana-5209	365	2	129	129	NUM
cana-5209	365	3	,	,	PUNCT
cana-5209	365	4	1	1	NUM
cana-5209	365	5	(	(	PUNCT
cana-5209	365	6	2006	2006	NUM
cana-5209	365	7	)	)	PUNCT
cana-5209	365	8	185199	185199	NUM
cana-5209	365	9	.	.	PUNCT
cana-5209	366	1	[	[	X
cana-5209	366	2	14	14	NUM
cana-5209	366	3	]	]	X
cana-5209	366	4	g.	g.	PROPN
cana-5209	366	5	caristi	caristi	PROPN
cana-5209	366	6	,	,	PUNCT
cana-5209	366	7	m.	m.	NOUN
cana-5209	366	8	ferrara	ferrara	NOUN
cana-5209	366	9	and	and	CCONJ
cana-5209	366	10	a.	a.	NOUN
cana-5209	366	11	stefanescu	stefanescu	PROPN
cana-5209	366	12	,	,	PUNCT
cana-5209	366	13	semi	semi	ADJ
cana-5209	366	14	-	-	ADJ
cana-5209	366	15	infinite	infinite	ADJ
cana-5209	366	16	multiobjective	multiobjective	ADJ
cana-5209	366	17	programming	programming	NOUN
cana-5209	366	18	with	with	ADP
cana-5209	366	19	generalized	generalized	ADJ
cana-5209	366	20	invexity	invexity	NOUN
cana-5209	366	21	,	,	PUNCT
cana-5209	366	22	math	math	NOUN
cana-5209	366	23	reports	report	NOUN
cana-5209	366	24	,	,	PUNCT
cana-5209	366	25	3	3	NUM
cana-5209	366	26	(	(	PUNCT
cana-5209	366	27	2010	2010	NUM
cana-5209	366	28	)	)	PUNCT
cana-5209	366	29	217	217	NUM
cana-5209	366	30	-	-	SYM
cana-5209	366	31	233	233	NUM
cana-5209	366	32	.	.	PUNCT
cana-5209	367	1	[	[	X
cana-5209	367	2	15	15	NUM
cana-5209	367	3	]	]	X
cana-5209	367	4	h.	h.	PROPN
cana-5209	367	5	kuk	kuk	PROPN
cana-5209	367	6	,	,	PUNCT
cana-5209	367	7	g.m	g.m	PROPN
cana-5209	367	8	.	.	PROPN
cana-5209	367	9	lee	lee	PROPN
cana-5209	367	10	and	and	CCONJ
cana-5209	367	11	t.	t.	PROPN
cana-5209	367	12	tanino	tanino	PROPN
cana-5209	367	13	,	,	PUNCT
cana-5209	367	14	optimality	optimality	NOUN
cana-5209	367	15	and	and	CCONJ
cana-5209	367	16	duality	duality	NOUN
cana-5209	367	17	for	for	ADP
cana-5209	367	18	nonsmooth	nonsmooth	ADJ
cana-5209	367	19	multiobjective	multiobjective	ADJ
cana-5209	367	20	fractional	fractional	ADJ
cana-5209	367	21	programming	programming	NOUN
cana-5209	367	22	with	with	ADP
cana-5209	367	23	generalized	generalized	ADJ
cana-5209	367	24	invexity	invexity	NOUN
cana-5209	367	25	,	,	PUNCT
cana-5209	367	26	journal	journal	NOUN
cana-5209	367	27	of	of	ADP
cana-5209	367	28	mathematical	mathematical	ADJ
cana-5209	367	29	analysis	analysis	NOUN
cana-5209	367	30	and	and	CCONJ
cana-5209	367	31	applications	application	NOUN
cana-5209	367	32	,	,	PUNCT
cana-5209	367	33	262	262	NUM
cana-5209	367	34	(	(	PUNCT
cana-5209	367	35	2001	2001	NUM
cana-5209	367	36	)	)	PUNCT
cana-5209	367	37	365	365	NUM
cana-5209	367	38	-	-	SYM
cana-5209	367	39	375	375	NUM
cana-5209	367	40	.	.	PUNCT
cana-5209	368	1	[	[	X
cana-5209	368	2	16	16	NUM
cana-5209	368	3	]	]	X
cana-5209	368	4	j.p	j.p	PROPN
cana-5209	368	5	.	.	PROPN
cana-5209	368	6	vial	vial	ADJ
cana-5209	368	7	,	,	PUNCT
cana-5209	368	8	strong	strong	ADJ
cana-5209	368	9	and	and	CCONJ
cana-5209	368	10	weak	weak	ADJ
cana-5209	368	11	convexity	convexity	NOUN
cana-5209	368	12	of	of	ADP
cana-5209	368	13	sets	set	NOUN
cana-5209	368	14	and	and	CCONJ
cana-5209	368	15	functions	function	NOUN
cana-5209	368	16	,	,	PUNCT
cana-5209	368	17	math	math	NOUN
cana-5209	368	18	.	.	PUNCT
cana-5209	369	1	oper	oper	PROPN
cana-5209	369	2	.	.	PUNCT
cana-5209	369	3	res	res	PROPN
cana-5209	369	4	.	.	PROPN
cana-5209	369	5	,	,	PUNCT
cana-5209	369	6	8	8	NUM
cana-5209	369	7	(	(	PUNCT
cana-5209	369	8	1983	1983	NUM
cana-5209	369	9	)	)	PUNCT
cana-5209	369	10	231	231	NUM
cana-5209	369	11	-	-	SYM
cana-5209	369	12	259	259	NUM
cana-5209	369	13	.	.	PUNCT
cana-5209	370	1	[	[	X
cana-5209	370	2	17	17	NUM
cana-5209	370	3	]	]	X
cana-5209	370	4	m.a	m.a	PROPN
cana-5209	370	5	.	.	PROPN
cana-5209	370	6	hanson	hanson	PROPN
cana-5209	370	7	,	,	PUNCT
cana-5209	370	8	on	on	ADP
cana-5209	370	9	sufficiency	sufficiency	NOUN
cana-5209	370	10	of	of	ADP
cana-5209	370	11	kuhn	kuhn	PROPN
cana-5209	370	12	tucker	tucker	PROPN
cana-5209	370	13	conditions	conditions	PROPN
cana-5209	370	14	,	,	PUNCT
cana-5209	370	15	j.	j.	PROPN
cana-5209	370	16	math	math	PROPN
cana-5209	370	17	.	.	PUNCT
cana-5209	371	1	anal	anal	PROPN
cana-5209	371	2	.	.	PUNCT
cana-5209	372	1	appl	appl	PROPN
cana-5209	372	2	.	.	PROPN
cana-5209	372	3	,	,	PUNCT
cana-5209	372	4	30	30	NUM
cana-5209	372	5	(	(	PUNCT
cana-5209	372	6	1981	1981	NUM
cana-5209	372	7	)	)	PUNCT
cana-5209	372	8	545550	545550	NUM
cana-5209	372	9	.	.	PUNCT
cana-5209	373	1	[	[	X
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cana-5209	373	3	]	]	X
cana-5209	373	4	m.a	m.a	PROPN
cana-5209	373	5	.	.	PROPN
cana-5209	373	6	hanson	hanson	PROPN
cana-5209	373	7	and	and	CCONJ
cana-5209	373	8	b.	b.	PROPN
cana-5209	373	9	mond	mond	PROPN
cana-5209	373	10	,	,	PUNCT
cana-5209	373	11	further	further	ADJ
cana-5209	373	12	generalizations	generalization	NOUN
cana-5209	373	13	of	of	ADP
cana-5209	373	14	convexity	convexity	NOUN
cana-5209	373	15	in	in	ADP
cana-5209	373	16	mathematical	mathematical	ADJ
cana-5209	373	17	programming	programming	NOUN
cana-5209	373	18	,	,	PUNCT
cana-5209	373	19	j.	j.	PROPN
cana-5209	373	20	inf	inf	PROPN
cana-5209	373	21	.	.	PROPN
cana-5209	373	22	optim	optim	PROPN
cana-5209	373	23	.	.	PUNCT
cana-5209	374	1	sci	sci	PROPN
cana-5209	374	2	.	.	PROPN
cana-5209	374	3	,	,	PUNCT
cana-5209	374	4	3	3	NUM
cana-5209	374	5	(	(	PUNCT
cana-5209	374	6	1982	1982	NUM
cana-5209	374	7	)	)	PUNCT
cana-5209	374	8	25	25	NUM
cana-5209	374	9	-	-	SYM
cana-5209	374	10	32	32	NUM
cana-5209	374	11	.	.	PUNCT
cana-5209	375	1	[	[	X
cana-5209	375	2	19	19	NUM
cana-5209	375	3	]	]	PUNCT
cana-5209	375	4	m.	m.	NOUN
cana-5209	375	5	jaiswal	jaiswal	PROPN
cana-5209	375	6	and	and	CCONJ
cana-5209	375	7	s.k	s.k	PROPN
cana-5209	375	8	.	.	PROPN
cana-5209	375	9	mishra	mishra	PROPN
cana-5209	375	10	,	,	PUNCT
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cana-5209	375	12	conditions	condition	NOUN
cana-5209	375	13	and	and	CCONJ
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cana-5209	375	15	for	for	ADP
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cana-5209	375	17	semi	semi	ADJ
cana-5209	375	18	-	-	ADJ
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cana-5209	375	24	(	(	PUNCT
cana-5209	375	25	c	c	X
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cana-5209	375	30	,	,	PUNCT
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cana-5209	375	32	,	,	PUNCT
cana-5209	375	33	ann	ann	PROPN
cana-5209	375	34	.	.	PROPN
cana-5209	375	35	univ	univ	PROPN
cana-5209	375	36	.	.	PUNCT
cana-5209	376	1	buchar	buchar	PROPN
cana-5209	376	2	.	.	PUNCT
cana-5209	377	1	math	math	PROPN
cana-5209	377	2	.	.	PUNCT
cana-5209	378	1	ser	ser	PROPN
cana-5209	378	2	.	.	PROPN
cana-5209	379	1	6(lxiv	6(lxiv	NUM
cana-5209	379	2	)	)	PUNCT
cana-5209	379	3	(	(	PUNCT
cana-5209	379	4	2015	2015	NUM
cana-5209	379	5	)	)	PUNCT
cana-5209	379	6	83	83	NUM
cana-5209	379	7	-	-	SYM
cana-5209	379	8	98	98	NUM
cana-5209	379	9	.	.	PUNCT
cana-5209	380	1	https://internationalpubls.com	https://internationalpubls.com	X
cana-5209	380	2	1150	1150	NUM
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cana-5209	380	4	on	on	ADP
cana-5209	380	5	applied	apply	VERB
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cana-5209	380	7	analysis	analysis	NOUN
cana-5209	380	8	issn	issn	NOUN
cana-5209	380	9	:	:	PUNCT
cana-5209	380	10	1074	1074	NUM
cana-5209	380	11	-	-	PUNCT
cana-5209	380	12	133x	133x	NUM
cana-5209	380	13	vol	vol	VERB
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cana-5209	380	16	.	.	PUNCT
cana-5209	381	1	10s	10	NOUN
cana-5209	381	2	(	(	PUNCT
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cana-5209	381	4	)	)	PUNCT
cana-5209	382	1	[	[	X
cana-5209	382	2	20	20	NUM
cana-5209	382	3	]	]	X
cana-5209	382	4	n.	n.	PROPN
cana-5209	382	5	kanzi	kanzi	PROPN
cana-5209	382	6	and	and	CCONJ
cana-5209	382	7	s.	s.	PROPN
cana-5209	382	8	nobakhtian	nobakhtian	PROPN
cana-5209	382	9	,	,	PUNCT
cana-5209	382	10	optimality	optimality	NOUN
cana-5209	382	11	conditions	condition	NOUN
cana-5209	382	12	for	for	ADP
cana-5209	382	13	nonsmooth	nonsmooth	ADJ
cana-5209	382	14	semi	semi	ADJ
cana-5209	382	15	-	-	ADJ
cana-5209	382	16	infnite	infnite	ADJ
cana-5209	382	17	programming	programming	NOUN
cana-5209	382	18	,	,	PUNCT
cana-5209	382	19	optimization	optimization	NOUN
cana-5209	382	20	,	,	PUNCT
cana-5209	382	21	59	59	NUM
cana-5209	382	22	(	(	PUNCT
cana-5209	382	23	2010	2010	NUM
cana-5209	382	24	)	)	PUNCT
cana-5209	382	25	717	717	NUM
cana-5209	382	26	-	-	SYM
cana-5209	382	27	727	727	NUM
cana-5209	382	28	.	.	PUNCT
cana-5209	383	1	[	[	X
cana-5209	383	2	21	21	NUM
cana-5209	383	3	]	]	X
cana-5209	383	4	s.k	s.k	PROPN
cana-5209	383	5	.	.	PROPN
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cana-5209	383	8	k.k	k.k	PROPN
cana-5209	383	9	.	.	PROPN
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cana-5209	383	14	,	,	PUNCT
cana-5209	383	15	optimality	optimality	NOUN
cana-5209	383	16	and	and	CCONJ
cana-5209	383	17	duality	duality	NOUN
cana-5209	383	18	for	for	ADP
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cana-5209	383	21	programming	programming	NOUN
cana-5209	383	22	with	with	ADP
cana-5209	383	23	support	support	NOUN
cana-5209	383	24	function	function	NOUN
cana-5209	383	25	under	under	ADP
cana-5209	383	26	(	(	PUNCT
cana-5209	383	27	c	c	X
cana-5209	383	28	,	,	PUNCT
cana-5209	383	29	α	α	PROPN
cana-5209	383	30	,	,	PUNCT
cana-5209	383	31	ρ	ρ	NOUN
cana-5209	383	32	,	,	PUNCT
cana-5209	383	33	d)-convexity	d)-convexity	NOUN
cana-5209	383	34	,	,	PUNCT
cana-5209	383	35	j.	j.	PROPN
cana-5209	383	36	comput	comput	PROPN
cana-5209	383	37	.	.	PUNCT
cana-5209	384	1	appl	appl	PROPN
cana-5209	384	2	.	.	PROPN
cana-5209	384	3	math	math	PROPN
cana-5209	384	4	.	.	PUNCT
cana-5209	385	1	274	274	NUM
cana-5209	385	2	(	(	PUNCT
cana-5209	385	3	2015	2015	NUM
cana-5209	385	4	)	)	PUNCT
cana-5209	385	5	1	1	NUM
cana-5209	385	6	-	-	SYM
cana-5209	385	7	10	10	NUM
cana-5209	385	8	.	.	PUNCT
cana-5209	386	1	[	[	X
cana-5209	386	2	22	22	NUM
cana-5209	386	3	]	]	X
cana-5209	386	4	s.k	s.k	PROPN
cana-5209	386	5	.	.	PROPN
cana-5209	386	6	mishra	mishra	PROPN
cana-5209	386	7	and	and	CCONJ
cana-5209	386	8	pooja	pooja	PROPN
cana-5209	386	9	gupta	gupta	PROPN
cana-5209	386	10	,	,	PUNCT
cana-5209	386	11	semiinfinite	semiinfinite	ADJ
cana-5209	386	12	multiobjective	multiobjective	ADJ
cana-5209	386	13	fractional	fractional	ADJ
cana-5209	386	14	programming	programming	NOUN
cana-5209	386	15	problems	problem	NOUN
cana-5209	386	16	involving	involve	VERB
cana-5209	386	17	type	type	NOUN
cana-5209	386	18	i	i	PRON
cana-5209	386	19	and	and	CCONJ
cana-5209	386	20	related	related	ADJ
cana-5209	386	21	functions	function	NOUN
cana-5209	386	22	,	,	PUNCT
cana-5209	386	23	investigations	investigation	NOUN
cana-5209	386	24	in	in	ADP
cana-5209	386	25	mathematical	mathematical	ADJ
cana-5209	386	26	sciences	science	NOUN
cana-5209	386	27	,	,	PUNCT
cana-5209	386	28	5	5	NUM
cana-5209	386	29	(	(	PUNCT
cana-5209	386	30	2016	2016	NUM
cana-5209	386	31	)	)	PUNCT
cana-5209	386	32	71	71	NUM
cana-5209	386	33	-	-	SYM
cana-5209	386	34	86	86	NUM
cana-5209	386	35	.	.	PUNCT
cana-5209	387	1	[	[	X
cana-5209	387	2	23	23	NUM
cana-5209	387	3	]	]	X
cana-5209	387	4	pooja	pooja	PROPN
cana-5209	387	5	gupta	gupta	PROPN
cana-5209	387	6	,	,	PUNCT
cana-5209	387	7	s.k	s.k	PROPN
cana-5209	387	8	.	.	PROPN
cana-5209	387	9	mishra	mishra	PROPN
cana-5209	387	10	and	and	CCONJ
cana-5209	387	11	r.n	r.n	PROPN
cana-5209	387	12	.	.	PROPN
cana-5209	387	13	mohapatra	mohapatra	PROPN
cana-5209	387	14	,	,	PUNCT
cana-5209	387	15	duality	duality	NOUN
cana-5209	387	16	models	model	NOUN
cana-5209	387	17	for	for	ADP
cana-5209	387	18	multiobjective	multiobjective	ADJ
cana-5209	387	19	semiinfinite	semiinfinite	ADJ
cana-5209	387	20	fractional	fractional	ADJ
cana-5209	387	21	programming	programming	NOUN
cana-5209	387	22	problems	problem	NOUN
cana-5209	387	23	involving	involve	VERB
cana-5209	387	24	type	type	NOUN
cana-5209	387	25	i	i	PRON
cana-5209	387	26	and	and	CCONJ
cana-5209	387	27	related	related	ADJ
cana-5209	387	28	functions	function	NOUN
cana-5209	387	29	,	,	PUNCT
cana-5209	387	30	quaestiones	quaestione	NOUN
cana-5209	387	31	mathematicae	mathematicae	PROPN
cana-5209	387	32	,	,	PUNCT
cana-5209	387	33	42	42	NUM
cana-5209	387	34	(	(	PUNCT
cana-5209	387	35	2018	2018	NUM
cana-5209	387	36	)	)	PUNCT
cana-5209	387	37	1199	1199	NUM
cana-5209	387	38	-	-	SYM
cana-5209	387	39	1220	1220	NUM
cana-5209	387	40	.	.	PUNCT
cana-5209	388	1	[	[	X
cana-5209	388	2	24	24	NUM
cana-5209	388	3	]	]	X
cana-5209	388	4	s.k	s.k	PROPN
cana-5209	388	5	.	.	PROPN
cana-5209	388	6	mishra	mishra	PROPN
cana-5209	388	7	,	,	PUNCT
cana-5209	388	8	m.	m.	PROPN
cana-5209	388	9	jaiswal	jaiswal	PROPN
cana-5209	388	10	and	and	CCONJ
cana-5209	388	11	l.t	l.t	PROPN
cana-5209	388	12	.	.	PROPN
cana-5209	388	13	hoai	hoai	PROPN
cana-5209	388	14	an	an	PRON
cana-5209	388	15	,	,	PUNCT
cana-5209	388	16	duality	duality	NOUN
cana-5209	388	17	for	for	ADP
cana-5209	388	18	nonsmooth	nonsmooth	ADJ
cana-5209	388	19	semi	semi	ADJ
cana-5209	388	20	-	-	ADJ
cana-5209	388	21	infnite	infnite	ADJ
cana-5209	388	22	programming	programming	NOUN
cana-5209	388	23	problems	problem	NOUN
cana-5209	388	24	,	,	PUNCT
cana-5209	388	25	optim	optim	ADJ
cana-5209	388	26	.	.	PUNCT
cana-5209	389	1	lett	lett	PROPN
cana-5209	389	2	.	.	PROPN
cana-5209	389	3	,	,	PUNCT
cana-5209	389	4	6	6	NUM
cana-5209	389	5	(	(	PUNCT
cana-5209	389	6	2012	2012	NUM
cana-5209	389	7	)	)	PUNCT
cana-5209	389	8	261	261	NUM
cana-5209	389	9	-	-	SYM
cana-5209	389	10	271	271	NUM
cana-5209	389	11	.	.	PUNCT
cana-5209	390	1	[	[	X
cana-5209	390	2	25	25	NUM
cana-5209	390	3	]	]	X
cana-5209	390	4	s.k	s.k	PROPN
cana-5209	390	5	.	.	PROPN
cana-5209	390	6	mishra	mishra	PROPN
cana-5209	390	7	,	,	PUNCT
cana-5209	390	8	m.	m.	PROPN
cana-5209	390	9	jaiswal	jaiswal	PROPN
cana-5209	390	10	and	and	CCONJ
cana-5209	390	11	l.t	l.t	PROPN
cana-5209	390	12	.	.	PROPN
cana-5209	390	13	hoai	hoai	PROPN
cana-5209	390	14	an	an	DET
cana-5209	390	15	,	,	PUNCT
cana-5209	390	16	optimality	optimality	NOUN
cana-5209	390	17	conditions	condition	NOUN
cana-5209	390	18	and	and	CCONJ
cana-5209	390	19	duality	duality	NOUN
cana-5209	390	20	for	for	ADP
cana-5209	390	21	nondifferentiable	nondifferentiable	ADJ
cana-5209	390	22	multiobjective	multiobjective	ADJ
cana-5209	390	23	semi	semi	ADJ
cana-5209	390	24	-	-	ADJ
cana-5209	390	25	infinite	infinite	ADJ
cana-5209	390	26	programming	programming	NOUN
cana-5209	390	27	problems	problem	NOUN
cana-5209	390	28	with	with	ADP
cana-5209	390	29	generalized	generalized	ADJ
cana-5209	390	30	(	(	PUNCT
cana-5209	390	31	c	c	X
cana-5209	390	32	,	,	PUNCT
cana-5209	390	33	α	α	PROPN
cana-5209	390	34	,	,	PUNCT
cana-5209	390	35	ρ	ρ	NOUN
cana-5209	390	36	,	,	PUNCT
cana-5209	390	37	d)-convexity	d)-convexity	NOUN
cana-5209	390	38	,	,	PUNCT
cana-5209	390	39	j	j	PROPN
cana-5209	390	40	syst	syst	PROPN
cana-5209	390	41	sci	sci	PROPN
cana-5209	390	42	complex	complex	PROPN
cana-5209	390	43	,	,	PUNCT
cana-5209	390	44	28	28	NUM
cana-5209	390	45	(	(	PUNCT
cana-5209	390	46	2015	2015	NUM
cana-5209	390	47	)	)	PUNCT
cana-5209	390	48	47	47	NUM
cana-5209	390	49	-	-	SYM
cana-5209	390	50	59	59	NUM
cana-5209	390	51	.	.	PUNCT
cana-5209	391	1	[	[	X
cana-5209	391	2	26	26	NUM
cana-5209	391	3	]	]	PUNCT
cana-5209	391	4	t.	t.	PROPN
cana-5209	391	5	maeda	maeda	PROPN
cana-5209	391	6	,	,	PUNCT
cana-5209	391	7	constraint	constraint	NOUN
cana-5209	391	8	qualification	qualification	NOUN
cana-5209	391	9	in	in	ADP
cana-5209	391	10	multiobjective	multiobjective	ADJ
cana-5209	391	11	problems	problem	NOUN
cana-5209	391	12	:	:	PUNCT
cana-5209	391	13	differentiable	differentiable	ADJ
cana-5209	391	14	case	case	NOUN
cana-5209	391	15	,	,	PUNCT
cana-5209	391	16	j.	j.	PROPN
cana-5209	391	17	optim	optim	PROPN
cana-5209	391	18	.	.	PUNCT
cana-5209	391	19	theory	theory	NOUN
cana-5209	391	20	appl	appl	PROPN
cana-5209	391	21	.	.	PROPN
cana-5209	391	22	,	,	PUNCT
cana-5209	391	23	80	80	NUM
cana-5209	391	24	(	(	PUNCT
cana-5209	391	25	1994	1994	NUM
cana-5209	391	26	)	)	PUNCT
cana-5209	391	27	483	483	NUM
cana-5209	391	28	-	-	SYM
cana-5209	391	29	500	500	NUM
cana-5209	391	30	.	.	PUNCT
cana-5209	392	1	[	[	X
cana-5209	392	2	27	27	NUM
cana-5209	392	3	]	]	X
cana-5209	392	4	t.	t.	PROPN
cana-5209	392	5	weir	weir	PROPN
cana-5209	392	6	,	,	PUNCT
cana-5209	392	7	a	a	DET
cana-5209	392	8	note	note	NOUN
cana-5209	392	9	on	on	ADP
cana-5209	392	10	invex	invex	NOUN
cana-5209	392	11	functions	function	NOUN
cana-5209	392	12	and	and	CCONJ
cana-5209	392	13	duality	duality	NOUN
cana-5209	392	14	in	in	ADP
cana-5209	392	15	multiple	multiple	ADJ
cana-5209	392	16	-	-	PUNCT
cana-5209	392	17	objective	objective	ADJ
cana-5209	392	18	optimization	optimization	NOUN
cana-5209	392	19	,	,	PUNCT
cana-5209	392	20	opsearch	opsearch	PROPN
cana-5209	392	21	,	,	PUNCT
cana-5209	392	22	25	25	NUM
cana-5209	392	23	(	(	PUNCT
cana-5209	392	24	1988	1988	NUM
cana-5209	392	25	)	)	PUNCT
cana-5209	392	26	98	98	NUM
cana-5209	392	27	-	-	SYM
cana-5209	392	28	104	104	NUM
cana-5209	392	29	.	.	PUNCT
cana-5209	393	1	[	[	X
cana-5209	393	2	28	28	NUM
cana-5209	393	3	]	]	X
cana-5209	393	4	v.	v.	X
cana-5209	393	5	preda	preda	PROPN
cana-5209	393	6	,	,	PUNCT
cana-5209	393	7	on	on	ADP
cana-5209	393	8	efficiency	efficiency	NOUN
cana-5209	393	9	and	and	CCONJ
cana-5209	393	10	duality	duality	NOUN
cana-5209	393	11	for	for	ADP
cana-5209	393	12	multiobjective	multiobjective	ADJ
cana-5209	393	13	programs	program	NOUN
cana-5209	393	14	,	,	PUNCT
cana-5209	393	15	j.	j.	PROPN
cana-5209	393	16	math	math	PROPN
cana-5209	393	17	.	.	PUNCT
cana-5209	394	1	anal	anal	PROPN
cana-5209	394	2	.	.	PUNCT
cana-5209	395	1	appl	appl	PROPN
cana-5209	395	2	.	.	PROPN
cana-5209	395	3	,	,	PUNCT
cana-5209	395	4	166	166	NUM
cana-5209	395	5	(	(	PUNCT
cana-5209	395	6	1992	1992	NUM
cana-5209	395	7	)	)	PUNCT
cana-5209	395	8	365	365	NUM
cana-5209	395	9	-	-	SYM
cana-5209	395	10	377	377	NUM
cana-5209	395	11	.	.	PUNCT
cana-5209	396	1	[	[	X
cana-5209	396	2	29	29	NUM
cana-5209	396	3	]	]	X
cana-5209	396	4	x.j	x.j	PROPN
cana-5209	396	5	.	.	PROPN
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cana-5209	396	7	,	,	PUNCT
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cana-5209	396	9	conditions	condition	NOUN
cana-5209	396	10	and	and	CCONJ
cana-5209	396	11	duality	duality	NOUN
cana-5209	396	12	for	for	ADP
cana-5209	396	13	nondifferentiable	nondifferentiable	ADJ
cana-5209	396	14	multiobjective	multiobjective	ADJ
cana-5209	396	15	fractional	fractional	ADJ
cana-5209	396	16	programming	programming	NOUN
cana-5209	396	17	problems	problem	NOUN
cana-5209	396	18	with	with	ADP
cana-5209	396	19	(	(	PUNCT
cana-5209	396	20	c	c	X
cana-5209	396	21	,	,	PUNCT
cana-5209	396	22	α	α	PROPN
cana-5209	396	23	,	,	PUNCT
cana-5209	396	24	ρ	ρ	NOUN
cana-5209	396	25	,	,	PUNCT
cana-5209	396	26	d)-convexity	d)-convexity	NOUN
cana-5209	396	27	,	,	PUNCT
cana-5209	396	28	j	j	PROPN
cana-5209	396	29	optim	optim	PROPN
cana-5209	396	30	theory	theory	NOUN
cana-5209	396	31	appl	appl	NOUN
cana-5209	396	32	,	,	PUNCT
cana-5209	396	33	148	148	NUM
cana-5209	396	34	(	(	PUNCT
cana-5209	396	35	2011	2011	NUM
cana-5209	396	36	)	)	PUNCT
cana-5209	396	37	197	197	NUM
cana-5209	396	38	-	-	SYM
cana-5209	396	39	208	208	NUM
cana-5209	396	40	.	.	PUNCT
cana-5209	397	1	[	[	X
cana-5209	397	2	30	30	NUM
cana-5209	397	3	]	]	X
cana-5209	397	4	z.a	z.a	PROPN
cana-5209	397	5	.	.	PUNCT
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cana-5209	397	8	h.x	h.x	PROPN
cana-5209	397	9	.	.	PROPN
cana-5209	397	10	huang	huang	PROPN
cana-5209	397	11	and	and	CCONJ
cana-5209	397	12	p.m.	p.m.	NOUN
cana-5209	397	13	pardalos	pardalo	NOUN
cana-5209	397	14	,	,	PUNCT
cana-5209	397	15	efficiency	efficiency	NOUN
cana-5209	397	16	conditions	condition	NOUN
cana-5209	397	17	and	and	CCONJ
cana-5209	397	18	duality	duality	NOUN
cana-5209	397	19	for	for	ADP
cana-5209	397	20	a	a	DET
cana-5209	397	21	class	class	NOUN
cana-5209	397	22	of	of	ADP
cana-5209	397	23	multiobjective	multiobjective	ADJ
cana-5209	397	24	fractional	fractional	ADJ
cana-5209	397	25	programming	programming	NOUN
cana-5209	397	26	problems	problem	NOUN
cana-5209	397	27	,	,	PUNCT
cana-5209	397	28	j.	j.	PROPN
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cana-5209	397	31	.	.	PROPN
cana-5209	397	32	,	,	PUNCT
cana-5209	397	33	27	27	NUM
cana-5209	397	34	(	(	PUNCT
cana-5209	397	35	2003	2003	NUM
cana-5209	397	36	)	)	PUNCT
cana-5209	397	37	447	447	NUM
cana-5209	397	38	-	-	SYM
cana-5209	397	39	471	471	NUM
cana-5209	397	40	.	.	PUNCT
cana-5209	398	1	[	[	X
cana-5209	398	2	31	31	NUM
cana-5209	398	3	]	]	X
cana-5209	398	4	z.a	z.a	PROPN
cana-5209	398	5	.	.	PROPN
cana-5209	398	6	liang	liang	PROPN
cana-5209	398	7	,	,	PUNCT
cana-5209	398	8	h.x	h.x	PROPN
cana-5209	398	9	.	.	PROPN
cana-5209	398	10	huang	huang	PROPN
cana-5209	398	11	and	and	CCONJ
cana-5209	398	12	p.m.	p.m.	NOUN
cana-5209	398	13	pardalos	pardalo	NOUN
cana-5209	398	14	,	,	PUNCT
cana-5209	398	15	optimality	optimality	NOUN
cana-5209	398	16	conditions	condition	NOUN
cana-5209	398	17	and	and	CCONJ
cana-5209	398	18	duality	duality	NOUN
cana-5209	398	19	for	for	ADP
cana-5209	398	20	a	a	DET
cana-5209	398	21	class	class	NOUN
cana-5209	398	22	of	of	ADP
cana-5209	398	23	nonlinear	nonlinear	ADJ
cana-5209	398	24	fractional	fractional	ADJ
cana-5209	398	25	programming	programming	NOUN
cana-5209	398	26	problems	problem	NOUN
cana-5209	398	27	,	,	PUNCT
cana-5209	398	28	j.	j.	PROPN
cana-5209	398	29	optim	optim	PROPN
cana-5209	398	30	.	.	PUNCT
cana-5209	399	1	theory	theory	NOUN
cana-5209	399	2	appl	appl	PROPN
cana-5209	399	3	.	.	PROPN
cana-5209	400	1	,	,	PUNCT
cana-5209	400	2	110	110	NUM
cana-5209	400	3	(	(	PUNCT
cana-5209	400	4	2001	2001	NUM
cana-5209	400	5	)	)	PUNCT
cana-5209	400	6	611	611	NUM
cana-5209	400	7	-	-	SYM
cana-5209	400	8	619	619	NUM
cana-5209	400	9	.	.	PUNCT
cana-5209	401	1	[	[	X
cana-5209	401	2	32	32	NUM
cana-5209	401	3	]	]	X
cana-5209	401	4	n.	n.	NOUN
cana-5209	401	5	abdulaleem	abdulaleem	NOUN
cana-5209	401	6	,	,	PUNCT
cana-5209	401	7	v	v	NOUN
cana-5209	401	8	-	-	PUNCT
cana-5209	401	9	e	e	NOUN
cana-5209	401	10	-	-	NOUN
cana-5209	401	11	invexity	invexity	NOUN
cana-5209	401	12	in	in	ADP
cana-5209	401	13	e	e	NOUN
cana-5209	401	14	-	-	ADJ
cana-5209	401	15	differentiable	differentiable	ADJ
cana-5209	401	16	multiobjective	multiobjective	ADJ
cana-5209	401	17	programming	programming	NOUN
cana-5209	401	18	,	,	PUNCT
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cana-5209	401	24	optimization	optimization	NOUN
cana-5209	401	25	,	,	PUNCT
cana-5209	401	26	12	12	NUM
cana-5209	401	27	(	(	PUNCT
cana-5209	401	28	2022	2022	NUM
cana-5209	401	29	)	)	PUNCT
cana-5209	401	30	427	427	NUM
cana-5209	401	31	-	-	SYM
cana-5209	401	32	443	443	NUM
cana-5209	401	33	.	.	PUNCT
cana-5209	402	1	[	[	X
cana-5209	402	2	33	33	NUM
cana-5209	402	3	]	]	PUNCT
cana-5209	402	4	t.	t.	PROPN
cana-5209	402	5	antczak	antczak	PROPN
cana-5209	402	6	and	and	CCONJ
cana-5209	402	7	a.	a.	NOUN
cana-5209	402	8	farajzadeh	farajzadeh	NOUN
cana-5209	402	9	,	,	PUNCT
cana-5209	402	10	on	on	ADP
cana-5209	402	11	nondifferentiable	nondifferentiable	ADJ
cana-5209	402	12	semi	semi	ADJ
cana-5209	402	13	-	-	ADJ
cana-5209	402	14	infinite	infinite	ADJ
cana-5209	402	15	multiobjective	multiobjective	ADJ
cana-5209	402	16	programming	programming	NOUN
cana-5209	402	17	with	with	ADP
cana-5209	402	18	interval	interval	NOUN
cana-5209	402	19	-	-	PUNCT
cana-5209	402	20	valued	value	VERB
cana-5209	402	21	functions	function	NOUN
cana-5209	402	22	,	,	PUNCT
cana-5209	402	23	journal	journal	NOUN
cana-5209	402	24	of	of	ADP
cana-5209	402	25	industrial	industrial	ADJ
cana-5209	402	26	and	and	CCONJ
cana-5209	402	27	management	management	NOUN
cana-5209	402	28	optimization	optimization	NOUN
cana-5209	402	29	,	,	PUNCT
cana-5209	402	30	19	19	NUM
cana-5209	402	31	(	(	PUNCT
cana-5209	402	32	2023	2023	NUM
cana-5209	402	33	)	)	PUNCT
cana-5209	402	34	5816	5816	NUM
cana-5209	402	35	-	-	SYM
cana-5209	402	36	5841	5841	NUM
cana-5209	402	37	.	.	PUNCT
cana-5209	403	1	[	[	X
cana-5209	403	2	34	34	NUM
cana-5209	403	3	]	]	PUNCT
cana-5209	403	4	s.	s.	PROPN
cana-5209	403	5	k.	k.	PROPN
cana-5209	403	6	mishra	mishra	PROPN
cana-5209	403	7	and	and	CCONJ
cana-5209	403	8	s.	s.	PROPN
cana-5209	403	9	k.	k.	PROPN
cana-5209	403	10	porwal	porwal	PROPN
cana-5209	403	11	,	,	PUNCT
cana-5209	403	12	on	on	ADP
cana-5209	403	13	sufficiency	sufficiency	NOUN
cana-5209	403	14	for	for	ADP
cana-5209	403	15	semiinfinite	semiinfinite	ADJ
cana-5209	403	16	multiobjective	multiobjective	ADJ
cana-5209	403	17	fractional	fractional	ADJ
cana-5209	403	18	programming	programming	NOUN
cana-5209	403	19	problems	problem	NOUN
cana-5209	403	20	using	use	VERB
cana-5209	403	21	generalized	generalized	ADJ
cana-5209	403	22	(	(	PUNCT
cana-5209	403	23	α	α	NOUN
cana-5209	403	24	,	,	PUNCT
cana-5209	403	25	η	η	PROPN
cana-5209	403	26	,	,	PUNCT
cana-5209	403	27	ρ	ρ	PROPN
cana-5209	403	28	)	)	PUNCT
cana-5209	403	29	−	−	PROPN
cana-5209	403	30	v−	v−	NOUN
cana-5209	403	31	univex	univex	ADJ
cana-5209	403	32	function	function	NOUN
cana-5209	403	33	,	,	PUNCT
cana-5209	403	34	communications	communication	NOUN
cana-5209	403	35	on	on	ADP
cana-5209	403	36	applied	apply	VERB
cana-5209	403	37	nonlinear	nonlinear	ADJ
cana-5209	403	38	analysis	analysis	NOUN
cana-5209	403	39	,	,	PUNCT
cana-5209	403	40	volume	volume	NOUN
cana-5209	403	41	22	22	NUM
cana-5209	403	42	,	,	PUNCT
cana-5209	403	43	issue	issue	NOUN
cana-5209	403	44	4	4	NUM
cana-5209	403	45	(	(	PUNCT
cana-5209	403	46	2015	2015	NUM
cana-5209	403	47	)	)	PUNCT
cana-5209	403	48	29	29	NUM
cana-5209	403	49	–	–	PUNCT
cana-5209	403	50	48	48	NUM
cana-5209	403	51	.	.	PUNCT
cana-5209	404	1	https://internationalpubls.com	https://internationalpubls.com	X
cana-5209	404	2	1151	1151	NUM