id	sid	tid	token	lemma	pos
ejpam-619	1	1	7_619_saini.dvi	7_619_saini.dvi	NUM
ejpam-619	1	2	european	european	PROPN
ejpam-619	1	3	journal	journal	PROPN
ejpam-619	1	4	of	of	ADP
ejpam-619	1	5	pure	pure	ADJ
ejpam-619	1	6	and	and	CCONJ
ejpam-619	1	7	applied	apply	VERB
ejpam-619	1	8	mathematics	mathematic	NOUN
ejpam-619	1	9	vol	vol	NOUN
ejpam-619	1	10	.	.	PROPN
ejpam-619	1	11	4	4	NUM
ejpam-619	1	12	,	,	PUNCT
ejpam-619	1	13	no	no	INTJ
ejpam-619	1	14	.	.	NOUN
ejpam-619	1	15	3	3	NUM
ejpam-619	1	16	,	,	PUNCT
ejpam-619	1	17	2011	2011	NUM
ejpam-619	1	18	,	,	PUNCT
ejpam-619	1	19	266	266	NUM
ejpam-619	1	20	-	-	SYM
ejpam-619	1	21	275	275	NUM
ejpam-619	1	22	issn	issn	PROPN
ejpam-619	1	23	1307	1307	NUM
ejpam-619	1	24	-	-	SYM
ejpam-619	1	25	5543	5543	NUM
ejpam-619	1	26	–	–	PUNCT
ejpam-619	1	27	www.ejpam.com	www.ejpam.com	X
ejpam-619	1	28	higher	high	ADJ
ejpam-619	1	29	-	-	PUNCT
ejpam-619	1	30	order	order	NOUN
ejpam-619	1	31	(	(	PUNCT
ejpam-619	1	32	f	f	X
ejpam-619	1	33	,	,	PUNCT
ejpam-619	1	34	α	α	PROPN
ejpam-619	1	35	,	,	PUNCT
ejpam-619	1	36	β	β	PROPN
ejpam-619	1	37	,	,	PUNCT
ejpam-619	1	38	ρ	ρ	PROPN
ejpam-619	1	39	,	,	PUNCT
ejpam-619	1	40	d)-convexity	d)-convexity	NOUN
ejpam-619	1	41	and	and	CCONJ
ejpam-619	1	42	its	its	PRON
ejpam-619	1	43	application	application	NOUN
ejpam-619	1	44	in	in	ADP
ejpam-619	1	45	fractional	fractional	ADJ
ejpam-619	1	46	programming	programming	NOUN
ejpam-619	1	47	t.	t.	PROPN
ejpam-619	1	48	r.	r.	PROPN
ejpam-619	1	49	gulati1,∗	gulati1,∗	PROPN
ejpam-619	1	50	,	,	PUNCT
ejpam-619	1	51	himani	himani	ADJ
ejpam-619	1	52	saini2	saini2	NOUN
ejpam-619	1	53	1	1	NUM
ejpam-619	1	54	department	department	NOUN
ejpam-619	1	55	of	of	ADP
ejpam-619	1	56	mathematics	mathematics	PROPN
ejpam-619	1	57	,	,	PUNCT
ejpam-619	1	58	professor	professor	NOUN
ejpam-619	1	59	,	,	PUNCT
ejpam-619	1	60	indian	indian	PROPN
ejpam-619	1	61	institute	institute	PROPN
ejpam-619	1	62	of	of	ADP
ejpam-619	1	63	technology	technology	PROPN
ejpam-619	1	64	,	,	PUNCT
ejpam-619	1	65	roorkee-247	roorkee-247	NOUN
ejpam-619	1	66	667	667	NUM
ejpam-619	1	67	,	,	PUNCT
ejpam-619	1	68	india	india	PROPN
ejpam-619	1	69	.	.	PROPN
ejpam-619	1	70	2	2	NUM
ejpam-619	1	71	applied	apply	VERB
ejpam-619	1	72	mathematics	mathematics	NOUN
ejpam-619	1	73	division	division	NOUN
ejpam-619	1	74	,	,	PUNCT
ejpam-619	1	75	scientist	scientist	NOUN
ejpam-619	1	76	,	,	PUNCT
ejpam-619	1	77	vikram	vikram	PROPN
ejpam-619	1	78	sarabhai	sarabhai	PROPN
ejpam-619	1	79	space	space	PROPN
ejpam-619	1	80	centre	centre	PROPN
ejpam-619	1	81	,	,	PUNCT
ejpam-619	1	82	indian	indian	ADJ
ejpam-619	1	83	space	space	NOUN
ejpam-619	1	84	research	research	NOUN
ejpam-619	1	85	organisation	organisation	NOUN
ejpam-619	1	86	,	,	PUNCT
ejpam-619	1	87	thiruvananthapuram-695	thiruvananthapuram-695	NOUN
ejpam-619	1	88	022	022	PROPN
ejpam-619	1	89	,	,	PUNCT
ejpam-619	1	90	india	india	PROPN
ejpam-619	1	91	.	.	PUNCT
ejpam-619	2	1	abstract	abstract	PROPN
ejpam-619	2	2	.	.	PUNCT
ejpam-619	3	1	in	in	ADP
ejpam-619	3	2	this	this	DET
ejpam-619	3	3	paper	paper	NOUN
ejpam-619	3	4	we	we	PRON
ejpam-619	3	5	introduce	introduce	VERB
ejpam-619	3	6	the	the	DET
ejpam-619	3	7	concept	concept	NOUN
ejpam-619	3	8	of	of	ADP
ejpam-619	3	9	higher	high	ADJ
ejpam-619	3	10	-	-	PUNCT
ejpam-619	3	11	order	order	NOUN
ejpam-619	3	12	(	(	PUNCT
ejpam-619	3	13	f	f	X
ejpam-619	3	14	,	,	PUNCT
ejpam-619	3	15	α	α	PROPN
ejpam-619	3	16	,	,	PUNCT
ejpam-619	3	17	β	β	PROPN
ejpam-619	3	18	,	,	PUNCT
ejpam-619	3	19	ρ	ρ	PROPN
ejpam-619	3	20	,	,	PUNCT
ejpam-619	3	21	d)-convexity	d)-convexity	NOUN
ejpam-619	3	22	with	with	ADP
ejpam-619	3	23	respect	respect	NOUN
ejpam-619	3	24	to	to	ADP
ejpam-619	3	25	a	a	DET
ejpam-619	3	26	differentiable	differentiable	ADJ
ejpam-619	3	27	function	function	NOUN
ejpam-619	3	28	k	k	PROPN
ejpam-619	3	29	.	.	PUNCT
ejpam-619	4	1	based	base	VERB
ejpam-619	4	2	on	on	ADP
ejpam-619	4	3	this	this	DET
ejpam-619	4	4	generalized	generalize	VERB
ejpam-619	4	5	convexity	convexity	NOUN
ejpam-619	4	6	,	,	PUNCT
ejpam-619	4	7	sufficient	sufficient	ADJ
ejpam-619	4	8	optimality	optimality	NOUN
ejpam-619	4	9	conditions	condition	NOUN
ejpam-619	4	10	for	for	ADP
ejpam-619	4	11	a	a	DET
ejpam-619	4	12	nonlinear	nonlinear	ADJ
ejpam-619	4	13	programming	programming	NOUN
ejpam-619	4	14	problem	problem	NOUN
ejpam-619	4	15	(	(	PUNCT
ejpam-619	4	16	np	np	X
ejpam-619	4	17	)	)	PUNCT
ejpam-619	4	18	are	be	AUX
ejpam-619	4	19	obtained	obtain	VERB
ejpam-619	4	20	.	.	PUNCT
ejpam-619	5	1	duality	duality	NOUN
ejpam-619	5	2	relations	relation	NOUN
ejpam-619	5	3	for	for	ADP
ejpam-619	5	4	mond	mond	NOUN
ejpam-619	5	5	-	-	PUNCT
ejpam-619	5	6	weir	weir	PROPN
ejpam-619	5	7	and	and	CCONJ
ejpam-619	5	8	wolfe	wolfe	PROPN
ejpam-619	5	9	duals	dual	NOUN
ejpam-619	5	10	of	of	ADP
ejpam-619	5	11	(	(	PUNCT
ejpam-619	5	12	np	np	INTJ
ejpam-619	5	13	)	)	PUNCT
ejpam-619	5	14	have	have	AUX
ejpam-619	5	15	also	also	ADV
ejpam-619	5	16	been	be	AUX
ejpam-619	5	17	discussed	discuss	VERB
ejpam-619	5	18	.	.	PUNCT
ejpam-619	6	1	these	these	DET
ejpam-619	6	2	duality	duality	NOUN
ejpam-619	6	3	results	result	NOUN
ejpam-619	6	4	are	be	AUX
ejpam-619	6	5	then	then	ADV
ejpam-619	6	6	applied	apply	VERB
ejpam-619	6	7	to	to	ADP
ejpam-619	6	8	nonlinear	nonlinear	ADJ
ejpam-619	6	9	fractional	fractional	ADJ
ejpam-619	6	10	programming	programming	NOUN
ejpam-619	6	11	problems	problem	NOUN
ejpam-619	6	12	.	.	PUNCT
ejpam-619	7	1	2000	2000	NUM
ejpam-619	7	2	mathematics	mathematic	NOUN
ejpam-619	7	3	subject	subject	NOUN
ejpam-619	7	4	classifications	classification	NOUN
ejpam-619	7	5	:	:	PUNCT
ejpam-619	7	6	90c30	90c30	NUM
ejpam-619	7	7	,	,	PUNCT
ejpam-619	7	8	90c32	90c32	NUM
ejpam-619	7	9	,	,	PUNCT
ejpam-619	7	10	90c46	90c46	NUM
ejpam-619	7	11	.	.	PUNCT
ejpam-619	8	1	key	key	ADJ
ejpam-619	8	2	words	word	NOUN
ejpam-619	8	3	and	and	CCONJ
ejpam-619	8	4	phrases	phrase	NOUN
ejpam-619	8	5	:	:	PUNCT
ejpam-619	8	6	higher	high	ADJ
ejpam-619	8	7	-	-	PUNCT
ejpam-619	8	8	order	order	NOUN
ejpam-619	8	9	(	(	PUNCT
ejpam-619	8	10	f	f	X
ejpam-619	8	11	,	,	PUNCT
ejpam-619	8	12	α	α	PROPN
ejpam-619	8	13	,	,	PUNCT
ejpam-619	8	14	β	β	PROPN
ejpam-619	8	15	,	,	PUNCT
ejpam-619	8	16	ρ	ρ	PROPN
ejpam-619	8	17	,	,	PUNCT
ejpam-619	8	18	d)-convexity	d)-convexity	NOUN
ejpam-619	8	19	;	;	PUNCT
ejpam-619	8	20	sufficiency	sufficiency	NOUN
ejpam-619	8	21	;	;	PUNCT
ejpam-619	8	22	optimality	optimality	NOUN
ejpam-619	8	23	conditions	condition	NOUN
ejpam-619	8	24	;	;	PUNCT
ejpam-619	8	25	duality	duality	NOUN
ejpam-619	8	26	;	;	PUNCT
ejpam-619	8	27	fractional	fractional	ADJ
ejpam-619	8	28	programming	programming	NOUN
ejpam-619	8	29	.	.	PUNCT
ejpam-619	9	1	1	1	X
ejpam-619	9	2	.	.	X
ejpam-619	9	3	introduction	introduction	NOUN
ejpam-619	9	4	optimality	optimality	NOUN
ejpam-619	9	5	conditions	condition	NOUN
ejpam-619	9	6	and	and	CCONJ
ejpam-619	9	7	duality	duality	NOUN
ejpam-619	9	8	in	in	ADP
ejpam-619	9	9	nonlinear	nonlinear	ADJ
ejpam-619	9	10	programming	programming	NOUN
ejpam-619	9	11	were	be	AUX
ejpam-619	9	12	first	first	ADV
ejpam-619	9	13	investigated	investigate	VERB
ejpam-619	9	14	under	under	ADP
ejpam-619	9	15	convexity	convexity	NOUN
ejpam-619	9	16	assumptions	assumption	NOUN
ejpam-619	9	17	.	.	PUNCT
ejpam-619	10	1	as	as	SCONJ
ejpam-619	10	2	they	they	PRON
ejpam-619	10	3	have	have	AUX
ejpam-619	10	4	played	play	VERB
ejpam-619	10	5	an	an	DET
ejpam-619	10	6	important	important	ADJ
ejpam-619	10	7	role	role	NOUN
ejpam-619	10	8	in	in	ADP
ejpam-619	10	9	the	the	DET
ejpam-619	10	10	development	development	NOUN
ejpam-619	10	11	of	of	ADP
ejpam-619	10	12	mathematical	mathematical	ADJ
ejpam-619	10	13	programming	programming	NOUN
ejpam-619	10	14	,	,	PUNCT
ejpam-619	10	15	several	several	ADJ
ejpam-619	10	16	authors	author	NOUN
ejpam-619	10	17	have	have	AUX
ejpam-619	10	18	generalized	generalize	VERB
ejpam-619	10	19	the	the	DET
ejpam-619	10	20	concept	concept	NOUN
ejpam-619	10	21	of	of	ADP
ejpam-619	10	22	convexity	convexity	NOUN
ejpam-619	10	23	under	under	ADP
ejpam-619	10	24	which	which	PRON
ejpam-619	10	25	sufficient	sufficient	ADJ
ejpam-619	10	26	optimality	optimality	NOUN
ejpam-619	10	27	conditions	condition	NOUN
ejpam-619	10	28	and	and	CCONJ
ejpam-619	10	29	duality	duality	NOUN
ejpam-619	10	30	theorems	theorem	NOUN
ejpam-619	10	31	holds	hold	VERB
ejpam-619	10	32	.	.	PUNCT
ejpam-619	11	1	hanson	hanson	PROPN
ejpam-619	12	1	[	[	X
ejpam-619	12	2	2	2	NUM
ejpam-619	12	3	]	]	PUNCT
ejpam-619	12	4	defined	define	VERB
ejpam-619	12	5	invex	invex	NOUN
ejpam-619	12	6	functions	function	NOUN
ejpam-619	12	7	.	.	PUNCT
ejpam-619	13	1	the	the	DET
ejpam-619	13	2	concept	concept	NOUN
ejpam-619	13	3	of	of	ADP
ejpam-619	13	4	(	(	PUNCT
ejpam-619	13	5	f	f	X
ejpam-619	13	6	,	,	PUNCT
ejpam-619	13	7	ρ)-convexity	ρ)-convexity	NOUN
ejpam-619	13	8	was	be	AUX
ejpam-619	13	9	introduced	introduce	VERB
ejpam-619	13	10	by	by	ADP
ejpam-619	13	11	pareda	pareda	NOUN
ejpam-619	13	12	[	[	X
ejpam-619	13	13	7	7	X
ejpam-619	13	14	]	]	PUNCT
ejpam-619	13	15	as	as	ADP
ejpam-619	13	16	an	an	DET
ejpam-619	13	17	extension	extension	NOUN
ejpam-619	13	18	of	of	ADP
ejpam-619	13	19	f	f	PROPN
ejpam-619	13	20	convexity	convexity	NOUN
ejpam-619	13	21	[	[	X
ejpam-619	13	22	3	3	NUM
ejpam-619	13	23	]	]	PUNCT
ejpam-619	13	24	and	and	CCONJ
ejpam-619	13	25	ρ	ρ	PROPN
ejpam-619	13	26	-	-	NOUN
ejpam-619	13	27	convexity	convexity	NOUN
ejpam-619	13	28	[	[	X
ejpam-619	13	29	8	8	NUM
ejpam-619	13	30	]	]	PUNCT
ejpam-619	13	31	.	.	PUNCT
ejpam-619	14	1	liang	liang	PROPN
ejpam-619	14	2	et	et	PROPN
ejpam-619	14	3	al	al	PROPN
ejpam-619	14	4	.	.	PUNCT
ejpam-619	15	1	[	[	X
ejpam-619	15	2	4	4	X
ejpam-619	15	3	]	]	PUNCT
ejpam-619	15	4	introduced	introduce	VERB
ejpam-619	15	5	a	a	DET
ejpam-619	15	6	unified	unified	ADJ
ejpam-619	15	7	formulation	formulation	NOUN
ejpam-619	15	8	of	of	ADP
ejpam-619	15	9	generalized	generalized	ADJ
ejpam-619	15	10	convexity	convexity	NOUN
ejpam-619	15	11	called	call	VERB
ejpam-619	15	12	(	(	PUNCT
ejpam-619	15	13	f	f	PROPN
ejpam-619	15	14	,	,	PUNCT
ejpam-619	15	15	α	α	PROPN
ejpam-619	15	16	,	,	PUNCT
ejpam-619	15	17	ρ	ρ	NOUN
ejpam-619	15	18	,	,	PUNCT
ejpam-619	15	19	d)-convexity	d)-convexity	NOUN
ejpam-619	15	20	and	and	CCONJ
ejpam-619	15	21	obtained	obtain	VERB
ejpam-619	15	22	some	some	DET
ejpam-619	15	23	optimality	optimality	NOUN
ejpam-619	15	24	conditions	condition	NOUN
ejpam-619	15	25	and	and	CCONJ
ejpam-619	15	26	duality	duality	NOUN
ejpam-619	15	27	results	result	NOUN
ejpam-619	15	28	for	for	ADP
ejpam-619	15	29	nonlinear	nonlinear	ADJ
ejpam-619	15	30	fractional	fractional	ADJ
ejpam-619	15	31	programming	programming	NOUN
ejpam-619	15	32	problems	problem	NOUN
ejpam-619	15	33	.	.	PUNCT
ejpam-619	16	1	recently	recently	ADV
ejpam-619	16	2	,	,	PUNCT
ejpam-619	16	3	yuan	yuan	PROPN
ejpam-619	16	4	et	et	PROPN
ejpam-619	16	5	al	al	PROPN
ejpam-619	16	6	.	.	PUNCT
ejpam-619	17	1	[	[	X
ejpam-619	17	2	9	9	NUM
ejpam-619	17	3	]	]	PUNCT
ejpam-619	17	4	introduced	introduce	VERB
ejpam-619	17	5	the	the	DET
ejpam-619	17	6	concept	concept	NOUN
ejpam-619	17	7	of	of	ADP
ejpam-619	17	8	(	(	PUNCT
ejpam-619	17	9	c	c	PROPN
ejpam-619	17	10	,	,	PUNCT
ejpam-619	17	11	α	α	PROPN
ejpam-619	17	12	,	,	PUNCT
ejpam-619	17	13	ρ	ρ	NOUN
ejpam-619	17	14	,	,	PUNCT
ejpam-619	17	15	d)-convexity	d)-convexity	NOUN
ejpam-619	17	16	which	which	PRON
ejpam-619	17	17	is	be	AUX
ejpam-619	17	18	the	the	DET
ejpam-619	17	19	generalization	generalization	NOUN
ejpam-619	17	20	of	of	ADP
ejpam-619	17	21	(	(	PUNCT
ejpam-619	17	22	f	f	X
ejpam-619	17	23	,	,	PUNCT
ejpam-619	17	24	α	α	PROPN
ejpam-619	17	25	,	,	PUNCT
ejpam-619	17	26	ρ	ρ	NOUN
ejpam-619	17	27	,	,	PUNCT
ejpam-619	17	28	d)-convexity	d)-convexity	NOUN
ejpam-619	17	29	,	,	PUNCT
ejpam-619	17	30	and	and	CCONJ
ejpam-619	17	31	proved	prove	VERB
ejpam-619	17	32	optimality	optimality	NOUN
ejpam-619	17	33	conditions	condition	NOUN
ejpam-619	17	34	and	and	CCONJ
ejpam-619	17	35	duality	duality	NOUN
ejpam-619	17	36	theorems	theorem	NOUN
ejpam-619	17	37	for	for	ADP
ejpam-619	17	38	∗corresponding	∗corresponde	VERB
ejpam-619	17	39	author	author	NOUN
ejpam-619	17	40	.	.	PUNCT
ejpam-619	18	1	email	email	NOUN
ejpam-619	18	2	addresses	address	NOUN
ejpam-619	18	3	:	:	PUNCT
ejpam-619	18	4	trgmaiitr	trgmaiitr	PROPN
ejpam-619	18	5	�	�	PROPN
ejpam-619	18	6	rediffmail	rediffmail	NOUN
ejpam-619	18	7	.	.	PUNCT
ejpam-619	19	1	om	om	PROPN
ejpam-619	19	2	(	(	PUNCT
ejpam-619	19	3	t.	t.	PROPN
ejpam-619	19	4	gulati	gulati	PROPN
ejpam-619	19	5	)	)	PUNCT
ejpam-619	19	6	,	,	PUNCT
ejpam-619	19	7	himanisaini.iitr	himanisaini.iitr	PROPN
ejpam-619	19	8	�	�	NOUN
ejpam-619	19	9	gmail	gmail	NOUN
ejpam-619	19	10	.	.	PUNCT
ejpam-619	20	1	om	om	PROPN
ejpam-619	20	2	(	(	PUNCT
ejpam-619	20	3	h.	h.	PROPN
ejpam-619	20	4	saini	saini	PROPN
ejpam-619	20	5	)	)	PUNCT
ejpam-619	20	6	http://www.ejpam.com	http://www.ejpam.com	X
ejpam-619	21	1	266	266	NUM
ejpam-619	21	2	c	c	X
ejpam-619	21	3	©	©	PROPN
ejpam-619	21	4	2011	2011	NUM
ejpam-619	21	5	ejpam	ejpam	VERB
ejpam-619	21	6	all	all	DET
ejpam-619	21	7	rights	right	NOUN
ejpam-619	21	8	reserved	reserve	VERB
ejpam-619	21	9	.	.	PUNCT
ejpam-619	22	1	t.	t.	PROPN
ejpam-619	22	2	gulati	gulati	PROPN
ejpam-619	22	3	,	,	PUNCT
ejpam-619	22	4	h.	h.	PROPN
ejpam-619	22	5	saini	saini	PROPN
ejpam-619	22	6	/	/	SYM
ejpam-619	22	7	eur	eur	PROPN
ejpam-619	22	8	.	.	PUNCT
ejpam-619	23	1	j.	j.	PROPN
ejpam-619	23	2	pure	pure	PROPN
ejpam-619	23	3	appl	appl	PROPN
ejpam-619	23	4	.	.	PROPN
ejpam-619	23	5	math	math	PROPN
ejpam-619	23	6	,	,	PUNCT
ejpam-619	23	7	4	4	NUM
ejpam-619	23	8	(	(	PUNCT
ejpam-619	23	9	2011	2011	NUM
ejpam-619	23	10	)	)	PUNCT
ejpam-619	23	11	,	,	PUNCT
ejpam-619	23	12	266	266	NUM
ejpam-619	23	13	-	-	SYM
ejpam-619	23	14	275	275	NUM
ejpam-619	23	15	267	267	NUM
ejpam-619	23	16	non	non	ADJ
ejpam-619	23	17	-	-	ADJ
ejpam-619	23	18	differentiable	differentiable	ADJ
ejpam-619	23	19	minimax	minimax	NOUN
ejpam-619	23	20	fractional	fractional	ADJ
ejpam-619	23	21	programming	programming	NOUN
ejpam-619	23	22	problems	problem	NOUN
ejpam-619	23	23	.	.	PUNCT
ejpam-619	24	1	this	this	DET
ejpam-619	24	2	paper	paper	NOUN
ejpam-619	24	3	is	be	AUX
ejpam-619	24	4	organized	organize	VERB
ejpam-619	24	5	as	as	SCONJ
ejpam-619	24	6	follows	follow	VERB
ejpam-619	24	7	.	.	PUNCT
ejpam-619	25	1	in	in	ADP
ejpam-619	25	2	section	section	NOUN
ejpam-619	25	3	2	2	NUM
ejpam-619	25	4	,	,	PUNCT
ejpam-619	25	5	we	we	PRON
ejpam-619	25	6	define	define	VERB
ejpam-619	25	7	higher	high	ADJ
ejpam-619	25	8	-	-	PUNCT
ejpam-619	25	9	order	order	NOUN
ejpam-619	25	10	(	(	PUNCT
ejpam-619	25	11	f	f	X
ejpam-619	25	12	,	,	PUNCT
ejpam-619	25	13	α	α	PROPN
ejpam-619	25	14	,	,	PUNCT
ejpam-619	25	15	β	β	PROPN
ejpam-619	25	16	,	,	PUNCT
ejpam-619	25	17	ρ	ρ	PROPN
ejpam-619	25	18	,	,	PUNCT
ejpam-619	25	19	d)convex	d)convex	NOUN
ejpam-619	25	20	functions	function	NOUN
ejpam-619	25	21	.	.	PUNCT
ejpam-619	26	1	under	under	ADP
ejpam-619	26	2	this	this	DET
ejpam-619	26	3	generalized	generalized	ADJ
ejpam-619	26	4	convexity	convexity	NOUN
ejpam-619	26	5	,	,	PUNCT
ejpam-619	26	6	we	we	PRON
ejpam-619	26	7	obtain	obtain	VERB
ejpam-619	26	8	sufficient	sufficient	ADJ
ejpam-619	26	9	optimality	optimality	NOUN
ejpam-619	26	10	conditions	condition	NOUN
ejpam-619	26	11	for	for	ADP
ejpam-619	26	12	a	a	DET
ejpam-619	26	13	nonlinear	nonlinear	ADJ
ejpam-619	26	14	programming	programming	NOUN
ejpam-619	26	15	problem	problem	NOUN
ejpam-619	26	16	(	(	PUNCT
ejpam-619	26	17	np	np	INTJ
ejpam-619	26	18	)	)	PUNCT
ejpam-619	26	19	in	in	ADP
ejpam-619	26	20	section	section	NOUN
ejpam-619	26	21	3	3	NUM
ejpam-619	26	22	.	.	PUNCT
ejpam-619	27	1	in	in	ADP
ejpam-619	27	2	section	section	NOUN
ejpam-619	27	3	4	4	NUM
ejpam-619	27	4	we	we	PRON
ejpam-619	27	5	establish	establish	VERB
ejpam-619	27	6	weak	weak	ADJ
ejpam-619	27	7	and	and	CCONJ
ejpam-619	27	8	strong	strong	ADJ
ejpam-619	27	9	duality	duality	NOUN
ejpam-619	27	10	for	for	ADP
ejpam-619	27	11	mond	mond	NOUN
ejpam-619	27	12	-	-	PUNCT
ejpam-619	27	13	weir	weir	PROPN
ejpam-619	27	14	dual	dual	ADJ
ejpam-619	27	15	program	program	NOUN
ejpam-619	27	16	for	for	ADP
ejpam-619	27	17	(	(	PUNCT
ejpam-619	27	18	np	np	INTJ
ejpam-619	27	19	)	)	PUNCT
ejpam-619	27	20	.	.	PUNCT
ejpam-619	28	1	an	an	DET
ejpam-619	28	2	application	application	NOUN
ejpam-619	28	3	for	for	ADP
ejpam-619	28	4	a	a	DET
ejpam-619	28	5	fractional	fractional	ADJ
ejpam-619	28	6	programming	programming	NOUN
ejpam-619	28	7	problem	problem	NOUN
ejpam-619	28	8	(	(	PUNCT
ejpam-619	28	9	fp	fp	X
ejpam-619	28	10	)	)	PUNCT
ejpam-619	28	11	has	have	AUX
ejpam-619	28	12	been	be	AUX
ejpam-619	28	13	discussed	discuss	VERB
ejpam-619	28	14	in	in	ADP
ejpam-619	28	15	section	section	NOUN
ejpam-619	28	16	5	5	NUM
ejpam-619	28	17	.	.	PUNCT
ejpam-619	29	1	in	in	ADP
ejpam-619	29	2	the	the	DET
ejpam-619	29	3	last	last	ADJ
ejpam-619	29	4	section	section	NOUN
ejpam-619	29	5	we	we	PRON
ejpam-619	29	6	present	present	VERB
ejpam-619	29	7	wolfe	wolfe	PROPN
ejpam-619	29	8	duality	duality	NOUN
ejpam-619	29	9	for	for	ADP
ejpam-619	29	10	(	(	PUNCT
ejpam-619	29	11	np	np	INTJ
ejpam-619	29	12	)	)	PUNCT
ejpam-619	29	13	and	and	CCONJ
ejpam-619	29	14	(	(	PUNCT
ejpam-619	29	15	fp	fp	NOUN
ejpam-619	29	16	)	)	PUNCT
ejpam-619	29	17	.	.	PUNCT
ejpam-619	30	1	2	2	X
ejpam-619	30	2	.	.	X
ejpam-619	30	3	definitions	definition	NOUN
ejpam-619	30	4	and	and	CCONJ
ejpam-619	30	5	preliminaries	preliminary	NOUN
ejpam-619	30	6	we	we	PRON
ejpam-619	30	7	consider	consider	VERB
ejpam-619	30	8	the	the	DET
ejpam-619	30	9	following	follow	VERB
ejpam-619	30	10	nonlinear	nonlinear	ADJ
ejpam-619	30	11	programming	programming	NOUN
ejpam-619	30	12	problem	problem	NOUN
ejpam-619	30	13	:	:	PUNCT
ejpam-619	30	14	(	(	PUNCT
ejpam-619	30	15	np	np	INTJ
ejpam-619	30	16	)	)	PUNCT
ejpam-619	30	17	minimize	minimize	VERB
ejpam-619	30	18	φ(x	φ(x	NOUN
ejpam-619	30	19	)	)	PUNCT
ejpam-619	30	20	,	,	PUNCT
ejpam-619	30	21	subject	subject	ADJ
ejpam-619	30	22	to	to	ADP
ejpam-619	30	23	h(x)≦	h(x)≦	NOUN
ejpam-619	30	24	0	0	NUM
ejpam-619	30	25	,	,	PUNCT
ejpam-619	30	26	x	x	X
ejpam-619	30	27	∈	∈	NOUN
ejpam-619	30	28	x	x	X
ejpam-619	30	29	,	,	PUNCT
ejpam-619	30	30	where	where	SCONJ
ejpam-619	30	31	x	x	PRON
ejpam-619	30	32	is	be	AUX
ejpam-619	30	33	an	an	DET
ejpam-619	30	34	open	open	ADJ
ejpam-619	30	35	subset	subset	NOUN
ejpam-619	30	36	of	of	ADP
ejpam-619	30	37	rn	rn	PROPN
ejpam-619	30	38	and	and	CCONJ
ejpam-619	30	39	the	the	DET
ejpam-619	30	40	functionsφ	functionsφ	NOUN
ejpam-619	30	41	:	:	PUNCT
ejpam-619	30	42	x	x	SYM
ejpam-619	30	43	7→	7→	NUM
ejpam-619	30	44	r	r	NOUN
ejpam-619	30	45	and	and	CCONJ
ejpam-619	30	46	h=	h=	NOUN
ejpam-619	30	47	(	(	PUNCT
ejpam-619	30	48	h1,h2	h1,h2	PROPN
ejpam-619	30	49	,	,	PUNCT
ejpam-619	30	50	.	.	PUNCT
ejpam-619	30	51	.	.	PUNCT
ejpam-619	30	52	.	.	PUNCT
ejpam-619	31	1	,	,	PUNCT
ejpam-619	31	2	hm	hm	INTJ
ejpam-619	31	3	)	)	PUNCT
ejpam-619	31	4	:	:	PUNCT
ejpam-619	32	1	x	x	X
ejpam-619	32	2	7→	7→	NUM
ejpam-619	32	3	rm	rm	NOUN
ejpam-619	32	4	are	be	AUX
ejpam-619	32	5	differentiable	differentiable	ADJ
ejpam-619	32	6	on	on	ADP
ejpam-619	32	7	x	x	X
ejpam-619	32	8	.	.	PUNCT
ejpam-619	33	1	let	let	VERB
ejpam-619	33	2	s	s	VERB
ejpam-619	33	3	=	=	PUNCT
ejpam-619	33	4	{	{	PUNCT
ejpam-619	33	5	x	x	SYM
ejpam-619	33	6	∈	∈	PROPN
ejpam-619	33	7	x	x	X
ejpam-619	33	8	:	:	PUNCT
ejpam-619	33	9	h(x)≦	h(x)≦	X
ejpam-619	33	10	0	0	NUM
ejpam-619	33	11	}	}	PUNCT
ejpam-619	33	12	denote	denote	VERB
ejpam-619	33	13	the	the	DET
ejpam-619	33	14	set	set	NOUN
ejpam-619	33	15	of	of	ADP
ejpam-619	33	16	all	all	DET
ejpam-619	33	17	feasible	feasible	ADJ
ejpam-619	33	18	solutions	solution	NOUN
ejpam-619	33	19	for	for	ADP
ejpam-619	33	20	(	(	PUNCT
ejpam-619	33	21	np	np	INTJ
ejpam-619	33	22	)	)	PUNCT
ejpam-619	33	23	.	.	PUNCT
ejpam-619	34	1	definition	definition	NOUN
ejpam-619	34	2	1	1	NUM
ejpam-619	34	3	.	.	PUNCT
ejpam-619	35	1	a	a	DET
ejpam-619	35	2	functional	functional	ADJ
ejpam-619	35	3	f	f	NOUN
ejpam-619	35	4	:	:	PUNCT
ejpam-619	35	5	x	x	SYM
ejpam-619	35	6	×	×	NOUN
ejpam-619	35	7	x	x	PUNCT
ejpam-619	35	8	×rn	×rn	NOUN
ejpam-619	35	9	7→	7→	NUM
ejpam-619	35	10	r	r	NOUN
ejpam-619	35	11	is	be	AUX
ejpam-619	35	12	said	say	VERB
ejpam-619	35	13	to	to	PART
ejpam-619	35	14	be	be	AUX
ejpam-619	35	15	sublinear	sublinear	NOUN
ejpam-619	35	16	in	in	ADP
ejpam-619	35	17	the	the	DET
ejpam-619	35	18	third	third	ADJ
ejpam-619	35	19	variable	variable	NOUN
ejpam-619	35	20	,	,	PUNCT
ejpam-619	35	21	if	if	SCONJ
ejpam-619	35	22	for	for	ADP
ejpam-619	35	23	all	all	DET
ejpam-619	35	24	x	x	SYM
ejpam-619	35	25	,	,	PUNCT
ejpam-619	35	26	x̄	x̄	NUM
ejpam-619	35	27	∈	∈	PROPN
ejpam-619	35	28	x	x	PUNCT
ejpam-619	35	29	,	,	PUNCT
ejpam-619	35	30	(	(	PUNCT
ejpam-619	35	31	i	i	NOUN
ejpam-619	35	32	)	)	PUNCT
ejpam-619	35	33	f(x	f(x	PROPN
ejpam-619	35	34	,	,	PUNCT
ejpam-619	35	35	x̄	x̄	NOUN
ejpam-619	35	36	;	;	PUNCT
ejpam-619	35	37	ξ1	ξ1	PROPN
ejpam-619	35	38	+	+	PROPN
ejpam-619	35	39	ξ2)≦	ξ2)≦	PROPN
ejpam-619	35	40	f(x	f(x	PROPN
ejpam-619	35	41	,	,	PUNCT
ejpam-619	35	42	x̄;ξ1	x̄;ξ1	PROPN
ejpam-619	35	43	)	)	PUNCT
ejpam-619	36	1	+	+	CCONJ
ejpam-619	36	2	f(x	f(x	PROPN
ejpam-619	36	3	,	,	PUNCT
ejpam-619	36	4	x̄	x̄	NOUN
ejpam-619	36	5	;	;	PUNCT
ejpam-619	36	6	ξ2	ξ2	ADJ
ejpam-619	36	7	)	)	PUNCT
ejpam-619	36	8	,	,	PUNCT
ejpam-619	36	9	for	for	ADP
ejpam-619	36	10	all	all	DET
ejpam-619	36	11	ξ1,ξ2	ξ1,ξ2	PROPN
ejpam-619	36	12	∈	∈	PROPN
ejpam-619	36	13	rn	rn	NOUN
ejpam-619	36	14	;	;	PUNCT
ejpam-619	36	15	and	and	CCONJ
ejpam-619	36	16	(	(	PUNCT
ejpam-619	36	17	ii	ii	NOUN
ejpam-619	36	18	)	)	PUNCT
ejpam-619	36	19	f(x	f(x	PROPN
ejpam-619	36	20	,	,	PUNCT
ejpam-619	36	21	x̄	x̄	NOUN
ejpam-619	36	22	;	;	PUNCT
ejpam-619	36	23	αa	αa	X
ejpam-619	36	24	)	)	PUNCT
ejpam-619	36	25	=	=	SYM
ejpam-619	36	26	αf(x	αf(x	NOUN
ejpam-619	36	27	,	,	PUNCT
ejpam-619	36	28	x̄	x̄	NOUN
ejpam-619	36	29	;	;	PUNCT
ejpam-619	36	30	a	a	X
ejpam-619	36	31	)	)	PUNCT
ejpam-619	36	32	for	for	ADP
ejpam-619	36	33	all	all	DET
ejpam-619	36	34	α	α	PROPN
ejpam-619	36	35	∈	∈	NOUN
ejpam-619	36	36	r+	r+	NOUN
ejpam-619	36	37	,	,	PUNCT
ejpam-619	36	38	and	and	CCONJ
ejpam-619	36	39	a	a	DET
ejpam-619	36	40	∈	∈	PROPN
ejpam-619	36	41	rn	rn	PROPN
ejpam-619	36	42	.	.	PROPN
ejpam-619	36	43	from	from	ADP
ejpam-619	36	44	(	(	PUNCT
ejpam-619	36	45	ii	ii	NOUN
ejpam-619	36	46	)	)	PUNCT
ejpam-619	36	47	,	,	PUNCT
ejpam-619	36	48	it	it	PRON
ejpam-619	36	49	is	be	AUX
ejpam-619	36	50	clear	clear	ADJ
ejpam-619	37	1	that	that	SCONJ
ejpam-619	37	2	f(x	f(x	PROPN
ejpam-619	37	3	,	,	PUNCT
ejpam-619	37	4	x̄	x̄	NOUN
ejpam-619	37	5	;	;	PUNCT
ejpam-619	37	6	0	0	X
ejpam-619	37	7	)	)	PUNCT
ejpam-619	37	8	=	=	SYM
ejpam-619	37	9	0	0	X
ejpam-619	37	10	.	.	PUNCT
ejpam-619	37	11	based	base	VERB
ejpam-619	37	12	on	on	ADP
ejpam-619	37	13	the	the	DET
ejpam-619	37	14	concept	concept	NOUN
ejpam-619	37	15	of	of	ADP
ejpam-619	37	16	the	the	DET
ejpam-619	37	17	sublinear	sublinear	NOUN
ejpam-619	37	18	functional	functional	ADJ
ejpam-619	37	19	,	,	PUNCT
ejpam-619	37	20	we	we	PRON
ejpam-619	37	21	now	now	ADV
ejpam-619	37	22	introduce	introduce	VERB
ejpam-619	37	23	the	the	DET
ejpam-619	37	24	class	class	NOUN
ejpam-619	37	25	of	of	ADP
ejpam-619	37	26	higherorder	higherorder	NOUN
ejpam-619	37	27	(	(	PUNCT
ejpam-619	37	28	f	f	X
ejpam-619	37	29	,	,	PUNCT
ejpam-619	37	30	α	α	PROPN
ejpam-619	37	31	,	,	PUNCT
ejpam-619	37	32	β	β	PROPN
ejpam-619	37	33	,	,	PUNCT
ejpam-619	37	34	ρ	ρ	PROPN
ejpam-619	37	35	,	,	PUNCT
ejpam-619	37	36	d)-convex	d)-convex	PUNCT
ejpam-619	37	37	functions	function	NOUN
ejpam-619	37	38	as	as	SCONJ
ejpam-619	37	39	follows	follow	VERB
ejpam-619	37	40	:	:	PUNCT
ejpam-619	37	41	let	let	VERB
ejpam-619	37	42	x	x	SYM
ejpam-619	37	43	⊆	⊆	NUM
ejpam-619	37	44	rn	rn	AUX
ejpam-619	37	45	be	be	AUX
ejpam-619	37	46	an	an	DET
ejpam-619	37	47	open	open	ADJ
ejpam-619	37	48	set	set	NOUN
ejpam-619	37	49	.	.	PUNCT
ejpam-619	38	1	let	let	VERB
ejpam-619	38	2	φ	φ	NOUN
ejpam-619	38	3	:	:	PUNCT
ejpam-619	38	4	x	x	SYM
ejpam-619	38	5	7→	7→	NUM
ejpam-619	38	6	r	r	NOUN
ejpam-619	38	7	,	,	PUNCT
ejpam-619	38	8	k	k	NOUN
ejpam-619	38	9	:	:	PUNCT
ejpam-619	38	10	x	x	SYM
ejpam-619	39	1	×	×	PROPN
ejpam-619	39	2	rn	rn	PROPN
ejpam-619	39	3	7→	7→	NUM
ejpam-619	39	4	r	r	NOUN
ejpam-619	39	5	be	be	VERB
ejpam-619	39	6	differentiable	differentiable	ADJ
ejpam-619	39	7	functions	function	NOUN
ejpam-619	39	8	,	,	PUNCT
ejpam-619	39	9	f	f	X
ejpam-619	39	10	:	:	PUNCT
ejpam-619	39	11	x	x	SYM
ejpam-619	39	12	×	×	NOUN
ejpam-619	39	13	x	x	PUNCT
ejpam-619	39	14	×rn	×rn	NOUN
ejpam-619	39	15	7→	7→	NUM
ejpam-619	39	16	r	r	NOUN
ejpam-619	39	17	be	be	VERB
ejpam-619	39	18	a	a	DET
ejpam-619	39	19	sublinear	sublinear	NOUN
ejpam-619	39	20	functional	functional	ADJ
ejpam-619	39	21	in	in	ADP
ejpam-619	39	22	the	the	DET
ejpam-619	39	23	third	third	ADJ
ejpam-619	39	24	variable	variable	NOUN
ejpam-619	39	25	and	and	CCONJ
ejpam-619	39	26	d	d	NOUN
ejpam-619	39	27	:	:	PUNCT
ejpam-619	39	28	x	x	SYM
ejpam-619	39	29	×	×	NOUN
ejpam-619	39	30	x	x	SYM
ejpam-619	39	31	7→	7→	PROPN
ejpam-619	39	32	r.	r.	PROPN
ejpam-619	39	33	further	far	ADV
ejpam-619	39	34	,	,	PUNCT
ejpam-619	39	35	let	let	VERB
ejpam-619	39	36	α	α	PRON
ejpam-619	39	37	,	,	PUNCT
ejpam-619	39	38	β	β	X
ejpam-619	39	39	:	:	PUNCT
ejpam-619	39	40	x	x	SYM
ejpam-619	39	41	×	×	NOUN
ejpam-619	39	42	x	x	SYM
ejpam-619	39	43	7→	7→	NUM
ejpam-619	39	44	r+	r+	PUNCT
ejpam-619	39	45	\	\	PUNCT
ejpam-619	39	46	{	{	PUNCT
ejpam-619	39	47	0	0	NUM
ejpam-619	39	48	}	}	PUNCT
ejpam-619	39	49	and	and	CCONJ
ejpam-619	39	50	ρ	ρ	PROPN
ejpam-619	39	51	∈	∈	PROPN
ejpam-619	39	52	r.	r.	PROPN
ejpam-619	39	53	definition	definition	NOUN
ejpam-619	39	54	2	2	NUM
ejpam-619	39	55	.	.	PUNCT
ejpam-619	40	1	the	the	DET
ejpam-619	40	2	function	function	NOUN
ejpam-619	40	3	φ	φ	PROPN
ejpam-619	40	4	is	be	AUX
ejpam-619	40	5	said	say	VERB
ejpam-619	40	6	to	to	PART
ejpam-619	40	7	be	be	AUX
ejpam-619	40	8	higher	high	ADJ
ejpam-619	40	9	-	-	PUNCT
ejpam-619	40	10	order	order	NOUN
ejpam-619	40	11	(	(	PUNCT
ejpam-619	40	12	f	f	X
ejpam-619	40	13	,	,	PUNCT
ejpam-619	40	14	α	α	PROPN
ejpam-619	40	15	,	,	PUNCT
ejpam-619	40	16	β	β	PROPN
ejpam-619	40	17	,	,	PUNCT
ejpam-619	40	18	ρ	ρ	PROPN
ejpam-619	40	19	,	,	PUNCT
ejpam-619	40	20	d)-convex	d)-convex	PUNCT
ejpam-619	40	21	at	at	ADP
ejpam-619	40	22	x̄	x̄	NOUN
ejpam-619	40	23	with	with	ADP
ejpam-619	40	24	respect	respect	NOUN
ejpam-619	40	25	to	to	ADP
ejpam-619	40	26	k	k	PROPN
ejpam-619	40	27	,	,	PUNCT
ejpam-619	40	28	if	if	SCONJ
ejpam-619	40	29	for	for	ADP
ejpam-619	40	30	all	all	DET
ejpam-619	40	31	x	x	SYM
ejpam-619	40	32	∈	∈	ADJ
ejpam-619	40	33	x	x	X
ejpam-619	40	34	and	and	CCONJ
ejpam-619	40	35	p	p	PROPN
ejpam-619	40	36	∈	∈	PROPN
ejpam-619	40	37	rn	rn	PROPN
ejpam-619	40	38	,	,	PUNCT
ejpam-619	40	39	φ(x)−φ	φ(x)−φ	PROPN
ejpam-619	40	40	(	(	PUNCT
ejpam-619	40	41	x̄	x̄	PROPN
ejpam-619	40	42	)	)	PUNCT
ejpam-619	40	43	≧	≧	X
ejpam-619	41	1	f(x	f(x	PROPN
ejpam-619	41	2	,	,	PUNCT
ejpam-619	41	3	x̄	x̄	PROPN
ejpam-619	41	4	;	;	PUNCT
ejpam-619	41	5	α(x	α(x	PROPN
ejpam-619	41	6	,	,	PUNCT
ejpam-619	41	7	x̄){∇φ	x̄){∇φ	PROPN
ejpam-619	41	8	(	(	PUNCT
ejpam-619	41	9	x̄	x̄	PROPN
ejpam-619	41	10	)	)	PUNCT
ejpam-619	42	1	+	+	X
ejpam-619	42	2	∇pk	∇pk	NOUN
ejpam-619	42	3	(	(	PUNCT
ejpam-619	42	4	x̄	x̄	NOUN
ejpam-619	42	5	,	,	PUNCT
ejpam-619	42	6	p	p	NOUN
ejpam-619	42	7	)	)	PUNCT
ejpam-619	42	8	}	}	PUNCT
ejpam-619	42	9	)	)	PUNCT
ejpam-619	42	10	+	+	NUM
ejpam-619	42	11	β(x	β(x	NOUN
ejpam-619	42	12	,	,	PUNCT
ejpam-619	42	13	x̄){k	x̄){k	NOUN
ejpam-619	42	14	(	(	PUNCT
ejpam-619	42	15	x̄	x̄	NOUN
ejpam-619	42	16	,	,	PUNCT
ejpam-619	42	17	p)−	p)−	PROPN
ejpam-619	42	18	pt∇pk	pt∇pk	PROPN
ejpam-619	42	19	(	(	PUNCT
ejpam-619	42	20	x̄	x̄	NOUN
ejpam-619	42	21	,	,	PUNCT
ejpam-619	42	22	p)}+ρd2(x	p)}+ρd2(x	PROPN
ejpam-619	42	23	,	,	PUNCT
ejpam-619	42	24	x̄	x̄	PROPN
ejpam-619	42	25	)	)	PUNCT
ejpam-619	42	26	.	.	PUNCT
ejpam-619	43	1	remark	remark	PROPN
ejpam-619	43	2	1	1	NUM
ejpam-619	43	3	.	.	PUNCT
ejpam-619	44	1	let	let	VERB
ejpam-619	44	2	k	k	X
ejpam-619	44	3	(	(	PUNCT
ejpam-619	44	4	x̄	x̄	NOUN
ejpam-619	44	5	,	,	PUNCT
ejpam-619	44	6	p	p	X
ejpam-619	44	7	)	)	PUNCT
ejpam-619	44	8	=	=	SYM
ejpam-619	45	1	0	0	X
ejpam-619	45	2	.	.	PUNCT
ejpam-619	46	1	(	(	PUNCT
ejpam-619	46	2	i	i	NOUN
ejpam-619	46	3	)	)	PUNCT
ejpam-619	46	4	then	then	ADV
ejpam-619	46	5	the	the	DET
ejpam-619	46	6	above	above	ADJ
ejpam-619	46	7	definition	definition	NOUN
ejpam-619	46	8	becomes	become	VERB
ejpam-619	46	9	that	that	PRON
ejpam-619	46	10	of	of	ADP
ejpam-619	46	11	(	(	PUNCT
ejpam-619	46	12	f	f	X
ejpam-619	46	13	,	,	PUNCT
ejpam-619	46	14	α	α	PROPN
ejpam-619	46	15	,	,	PUNCT
ejpam-619	46	16	ρ	ρ	NOUN
ejpam-619	46	17	,	,	PUNCT
ejpam-619	46	18	d)-convex	d)-convex	PUNCT
ejpam-619	46	19	function	function	NOUN
ejpam-619	46	20	introduced	introduce	VERB
ejpam-619	46	21	by	by	ADP
ejpam-619	46	22	liang	liang	PROPN
ejpam-619	46	23	et	et	PROPN
ejpam-619	46	24	al	al	PROPN
ejpam-619	46	25	.	.	PUNCT
ejpam-619	47	1	[	[	X
ejpam-619	47	2	4	4	NUM
ejpam-619	47	3	]	]	PUNCT
ejpam-619	47	4	.	.	PUNCT
ejpam-619	48	1	t.	t.	PROPN
ejpam-619	48	2	gulati	gulati	PROPN
ejpam-619	48	3	,	,	PUNCT
ejpam-619	48	4	h.	h.	PROPN
ejpam-619	48	5	saini	saini	PROPN
ejpam-619	48	6	/	/	SYM
ejpam-619	48	7	eur	eur	PROPN
ejpam-619	48	8	.	.	PUNCT
ejpam-619	49	1	j.	j.	PROPN
ejpam-619	49	2	pure	pure	PROPN
ejpam-619	49	3	appl	appl	PROPN
ejpam-619	49	4	.	.	PROPN
ejpam-619	49	5	math	math	PROPN
ejpam-619	49	6	,	,	PUNCT
ejpam-619	49	7	4	4	NUM
ejpam-619	49	8	(	(	PUNCT
ejpam-619	49	9	2011	2011	NUM
ejpam-619	49	10	)	)	PUNCT
ejpam-619	49	11	,	,	PUNCT
ejpam-619	49	12	266	266	NUM
ejpam-619	49	13	-	-	SYM
ejpam-619	49	14	275	275	NUM
ejpam-619	49	15	268	268	NUM
ejpam-619	49	16	(	(	PUNCT
ejpam-619	49	17	ii	ii	NOUN
ejpam-619	49	18	)	)	PUNCT
ejpam-619	49	19	if	if	SCONJ
ejpam-619	49	20	α(x	α(x	PROPN
ejpam-619	49	21	,	,	PUNCT
ejpam-619	49	22	x̄	x̄	PROPN
ejpam-619	49	23	)	)	PUNCT
ejpam-619	49	24	=	=	SYM
ejpam-619	50	1	1	1	NUM
ejpam-619	50	2	,	,	PUNCT
ejpam-619	50	3	we	we	PRON
ejpam-619	50	4	obtain	obtain	VERB
ejpam-619	50	5	the	the	DET
ejpam-619	50	6	definition	definition	NOUN
ejpam-619	50	7	of	of	ADP
ejpam-619	50	8	(	(	PUNCT
ejpam-619	50	9	f	f	X
ejpam-619	50	10	,	,	PUNCT
ejpam-619	50	11	ρ)-convex	ρ)-convex	NOUN
ejpam-619	50	12	function	function	NOUN
ejpam-619	50	13	given	give	VERB
ejpam-619	50	14	by	by	ADP
ejpam-619	50	15	pareda	pareda	NOUN
ejpam-619	50	16	[	[	X
ejpam-619	50	17	7	7	NUM
ejpam-619	50	18	]	]	PUNCT
ejpam-619	50	19	.	.	PUNCT
ejpam-619	51	1	(	(	PUNCT
ejpam-619	51	2	iii	iii	X
ejpam-619	51	3	)	)	PUNCT
ejpam-619	51	4	if	if	SCONJ
ejpam-619	51	5	α(x	α(x	PROPN
ejpam-619	51	6	,	,	PUNCT
ejpam-619	51	7	x̄	x̄	PROPN
ejpam-619	51	8	)	)	PUNCT
ejpam-619	51	9	=	=	SYM
ejpam-619	52	1	1,ρ	1,ρ	PROPN
ejpam-619	52	2	=	=	SYM
ejpam-619	52	3	0	0	PROPN
ejpam-619	52	4	and	and	CCONJ
ejpam-619	52	5	f(x	f(x	PROPN
ejpam-619	52	6	,	,	PUNCT
ejpam-619	52	7	x̄	x̄	NOUN
ejpam-619	52	8	;	;	PUNCT
ejpam-619	53	1	∇φ	∇φ	PROPN
ejpam-619	53	2	(	(	PUNCT
ejpam-619	53	3	x̄	x̄	NOUN
ejpam-619	53	4	)	)	PUNCT
ejpam-619	53	5	)	)	PUNCT
ejpam-619	54	1	=	=	PUNCT
ejpam-619	54	2	ηt	ηt	ADP
ejpam-619	54	3	(	(	PUNCT
ejpam-619	54	4	x	x	INTJ
ejpam-619	54	5	,	,	PUNCT
ejpam-619	54	6	x̄)∇φ	x̄)∇φ	PROPN
ejpam-619	54	7	(	(	PUNCT
ejpam-619	54	8	x̄	x̄	PROPN
ejpam-619	54	9	)	)	PUNCT
ejpam-619	54	10	for	for	ADP
ejpam-619	54	11	a	a	DET
ejpam-619	54	12	certain	certain	ADJ
ejpam-619	54	13	map	map	NOUN
ejpam-619	54	14	η	η	NOUN
ejpam-619	54	15	:	:	PUNCT
ejpam-619	54	16	x	x	SYM
ejpam-619	54	17	×	×	NOUN
ejpam-619	54	18	x	x	SYM
ejpam-619	54	19	−→	−→	PROPN
ejpam-619	54	20	rn	rn	PROPN
ejpam-619	54	21	,	,	PUNCT
ejpam-619	54	22	then	then	ADV
ejpam-619	54	23	(	(	PUNCT
ejpam-619	54	24	f	f	X
ejpam-619	54	25	,	,	PUNCT
ejpam-619	54	26	α	α	PROPN
ejpam-619	54	27	,	,	PUNCT
ejpam-619	54	28	β	β	PROPN
ejpam-619	54	29	,	,	PUNCT
ejpam-619	54	30	ρ	ρ	PROPN
ejpam-619	54	31	,	,	PUNCT
ejpam-619	54	32	d)-convexity	d)-convexity	NOUN
ejpam-619	54	33	reduces	reduce	VERB
ejpam-619	54	34	to	to	ADP
ejpam-619	54	35	the	the	DET
ejpam-619	54	36	invexity	invexity	NOUN
ejpam-619	54	37	in	in	ADP
ejpam-619	54	38	hanson	hanson	PROPN
ejpam-619	55	1	[	[	X
ejpam-619	55	2	2	2	NUM
ejpam-619	55	3	]	]	PUNCT
ejpam-619	55	4	.	.	PUNCT
ejpam-619	56	1	(	(	PUNCT
ejpam-619	56	2	iv	iv	X
ejpam-619	56	3	)	)	PUNCT
ejpam-619	56	4	if	if	SCONJ
ejpam-619	56	5	f	f	PROPN
ejpam-619	56	6	is	be	AUX
ejpam-619	56	7	convex	convex	ADJ
ejpam-619	56	8	with	with	ADP
ejpam-619	56	9	respect	respect	NOUN
ejpam-619	56	10	to	to	ADP
ejpam-619	56	11	the	the	DET
ejpam-619	56	12	third	third	ADJ
ejpam-619	56	13	argument	argument	NOUN
ejpam-619	56	14	,	,	PUNCT
ejpam-619	56	15	then	then	ADV
ejpam-619	56	16	we	we	PRON
ejpam-619	56	17	obtain	obtain	VERB
ejpam-619	56	18	the	the	DET
ejpam-619	56	19	definition	definition	NOUN
ejpam-619	56	20	of	of	ADP
ejpam-619	56	21	(	(	PUNCT
ejpam-619	56	22	f	f	X
ejpam-619	56	23	,	,	PUNCT
ejpam-619	56	24	α	α	PROPN
ejpam-619	56	25	,	,	PUNCT
ejpam-619	56	26	ρ	ρ	PROPN
ejpam-619	56	27	,	,	PUNCT
ejpam-619	56	28	d)convex	d)convex	PROPN
ejpam-619	56	29	function	function	NOUN
ejpam-619	56	30	introduced	introduce	VERB
ejpam-619	56	31	by	by	ADP
ejpam-619	56	32	yuan	yuan	PROPN
ejpam-619	56	33	et	et	PROPN
ejpam-619	56	34	al	al	PROPN
ejpam-619	56	35	.	.	PUNCT
ejpam-619	57	1	[	[	X
ejpam-619	57	2	9	9	NUM
ejpam-619	57	3	]	]	PUNCT
ejpam-619	57	4	.	.	PUNCT
ejpam-619	58	1	remark	remark	PROPN
ejpam-619	58	2	2	2	NUM
ejpam-619	58	3	.	.	PUNCT
ejpam-619	59	1	let	let	VERB
ejpam-619	59	2	β(x	β(x	NOUN
ejpam-619	59	3	,	,	PUNCT
ejpam-619	59	4	x̄	x̄	PROPN
ejpam-619	59	5	)	)	PUNCT
ejpam-619	60	1	=	=	SYM
ejpam-619	60	2	1	1	X
ejpam-619	60	3	.	.	PUNCT
ejpam-619	60	4	(	(	PUNCT
ejpam-619	60	5	i	i	NOUN
ejpam-619	60	6	)	)	PUNCT
ejpam-619	60	7	if	if	SCONJ
ejpam-619	60	8	k	k	X
ejpam-619	60	9	(	(	PUNCT
ejpam-619	60	10	x̄	x̄	NOUN
ejpam-619	60	11	,	,	PUNCT
ejpam-619	60	12	p	p	X
ejpam-619	60	13	)	)	PUNCT
ejpam-619	60	14	=	=	SYM
ejpam-619	60	15	1	1	NUM
ejpam-619	60	16	2	2	NUM
ejpam-619	60	17	pt∇2φ	pt∇2φ	NOUN
ejpam-619	60	18	(	(	PUNCT
ejpam-619	60	19	x̄)p	x̄)p	PROPN
ejpam-619	60	20	,	,	PUNCT
ejpam-619	60	21	then	then	ADV
ejpam-619	60	22	the	the	DET
ejpam-619	60	23	above	above	ADJ
ejpam-619	60	24	inequality	inequality	NOUN
ejpam-619	60	25	reduces	reduce	VERB
ejpam-619	60	26	to	to	ADP
ejpam-619	60	27	the	the	DET
ejpam-619	60	28	definition	definition	NOUN
ejpam-619	60	29	of	of	ADP
ejpam-619	60	30	second	second	ADJ
ejpam-619	60	31	order	order	NOUN
ejpam-619	60	32	(	(	PUNCT
ejpam-619	60	33	f	f	X
ejpam-619	60	34	,	,	PUNCT
ejpam-619	60	35	α	α	PROPN
ejpam-619	60	36	,	,	PUNCT
ejpam-619	60	37	ρ	ρ	NOUN
ejpam-619	60	38	,	,	PUNCT
ejpam-619	60	39	d)-convex	d)-convex	PUNCT
ejpam-619	60	40	function	function	NOUN
ejpam-619	60	41	given	give	VERB
ejpam-619	60	42	by	by	ADP
ejpam-619	60	43	ahmad	ahmad	NOUN
ejpam-619	60	44	and	and	CCONJ
ejpam-619	60	45	husain	husain	PROPN
ejpam-619	61	1	[	[	X
ejpam-619	61	2	1	1	NUM
ejpam-619	61	3	]	]	PUNCT
ejpam-619	61	4	.	.	PUNCT
ejpam-619	62	1	(	(	PUNCT
ejpam-619	62	2	ii	ii	NOUN
ejpam-619	62	3	)	)	PUNCT
ejpam-619	62	4	if	if	SCONJ
ejpam-619	62	5	α(x	α(x	PROPN
ejpam-619	62	6	,	,	PUNCT
ejpam-619	62	7	x̄	x̄	PROPN
ejpam-619	62	8	)	)	PUNCT
ejpam-619	62	9	=	=	SYM
ejpam-619	63	1	1,ρ	1,ρ	NUM
ejpam-619	63	2	=	=	SYM
ejpam-619	63	3	0	0	PROPN
ejpam-619	63	4	,	,	PUNCT
ejpam-619	63	5	k	k	X
ejpam-619	63	6	(	(	PUNCT
ejpam-619	63	7	x̄	x̄	NOUN
ejpam-619	63	8	,	,	PUNCT
ejpam-619	63	9	p	p	X
ejpam-619	63	10	)	)	PUNCT
ejpam-619	63	11	=	=	SYM
ejpam-619	63	12	1	1	NUM
ejpam-619	63	13	2	2	NUM
ejpam-619	63	14	pt∇2φ	pt∇2φ	NOUN
ejpam-619	63	15	(	(	PUNCT
ejpam-619	63	16	x̄)p	x̄)p	PROPN
ejpam-619	63	17	and	and	CCONJ
ejpam-619	63	18	f(x	f(x	PROPN
ejpam-619	63	19	,	,	PUNCT
ejpam-619	63	20	x̄	x̄	PROPN
ejpam-619	63	21	;	;	PUNCT
ejpam-619	63	22	a	a	X
ejpam-619	63	23	)	)	PUNCT
ejpam-619	63	24	=	=	NOUN
ejpam-619	63	25	ηt	ηt	ADP
ejpam-619	63	26	(	(	PUNCT
ejpam-619	63	27	x	x	INTJ
ejpam-619	63	28	,	,	PUNCT
ejpam-619	63	29	x̄)a	x̄)a	NOUN
ejpam-619	63	30	,	,	PUNCT
ejpam-619	63	31	where	where	SCONJ
ejpam-619	63	32	η	η	NOUN
ejpam-619	63	33	:	:	PUNCT
ejpam-619	63	34	x	x	SYM
ejpam-619	63	35	×	×	NOUN
ejpam-619	63	36	x	x	PUNCT
ejpam-619	63	37	−→	−→	PROPN
ejpam-619	63	38	rn	rn	PROPN
ejpam-619	63	39	,	,	PUNCT
ejpam-619	63	40	the	the	DET
ejpam-619	63	41	above	above	ADJ
ejpam-619	63	42	definition	definition	NOUN
ejpam-619	63	43	becomes	become	VERB
ejpam-619	63	44	that	that	PRON
ejpam-619	63	45	of	of	ADP
ejpam-619	63	46	η	η	PROPN
ejpam-619	63	47	-	-	PROPN
ejpam-619	63	48	bonvexity	bonvexity	NOUN
ejpam-619	63	49	introduced	introduce	VERB
ejpam-619	63	50	by	by	ADP
ejpam-619	63	51	pandey	pandey	PROPN
ejpam-619	64	1	[	[	X
ejpam-619	64	2	6	6	NUM
ejpam-619	64	3	]	]	PUNCT
ejpam-619	64	4	.	.	PUNCT
ejpam-619	65	1	proposition	proposition	NOUN
ejpam-619	65	2	1	1	NUM
ejpam-619	65	3	(	(	PUNCT
ejpam-619	65	4	kuhn	kuhn	PROPN
ejpam-619	65	5	-	-	PUNCT
ejpam-619	65	6	tucker	tucker	PROPN
ejpam-619	65	7	necessary	necessary	ADJ
ejpam-619	65	8	optimality	optimality	NOUN
ejpam-619	65	9	conditions	condition	NOUN
ejpam-619	65	10	[	[	PUNCT
ejpam-619	65	11	see	see	VERB
ejpam-619	65	12	5	5	NUM
ejpam-619	65	13	]	]	PUNCT
ejpam-619	65	14	)	)	PUNCT
ejpam-619	65	15	.	.	PUNCT
ejpam-619	66	1	let	let	VERB
ejpam-619	66	2	x̄	x̄	PRON
ejpam-619	66	3	∈	∈	PROPN
ejpam-619	66	4	s	s	AUX
ejpam-619	66	5	be	be	AUX
ejpam-619	66	6	an	an	DET
ejpam-619	66	7	optimal	optimal	ADJ
ejpam-619	66	8	solution	solution	NOUN
ejpam-619	66	9	of	of	ADP
ejpam-619	66	10	(	(	PUNCT
ejpam-619	66	11	np	np	INTJ
ejpam-619	66	12	)	)	PUNCT
ejpam-619	66	13	and	and	CCONJ
ejpam-619	66	14	let	let	VERB
ejpam-619	66	15	h	h	PRON
ejpam-619	66	16	satisfy	satisfy	VERB
ejpam-619	66	17	a	a	DET
ejpam-619	66	18	constraint	constraint	NOUN
ejpam-619	66	19	qualification	qualification	NOUN
ejpam-619	66	20	[	[	X
ejpam-619	66	21	theorem	theorem	VERB
ejpam-619	66	22	7.3.7	7.3.7	NOUN
ejpam-619	66	23	in	in	ADP
ejpam-619	66	24	5	5	NUM
ejpam-619	66	25	]	]	PUNCT
ejpam-619	66	26	.	.	PUNCT
ejpam-619	67	1	then	then	ADV
ejpam-619	67	2	there	there	PRON
ejpam-619	67	3	exists	exist	VERB
ejpam-619	67	4	a	a	DET
ejpam-619	67	5	v̄	v̄	NOUN
ejpam-619	67	6	∈	∈	PROPN
ejpam-619	67	7	rm	rm	NOUN
ejpam-619	67	8	such	such	ADJ
ejpam-619	67	9	that	that	SCONJ
ejpam-619	67	10	∇φ	∇φ	PROPN
ejpam-619	67	11	(	(	PUNCT
ejpam-619	67	12	x̄	x̄	PROPN
ejpam-619	67	13	)	)	PUNCT
ejpam-619	68	1	+	+	PUNCT
ejpam-619	68	2	∇h	∇h	PROPN
ejpam-619	68	3	(	(	PUNCT
ejpam-619	68	4	x̄)v̄	x̄)v̄	NOUN
ejpam-619	68	5	=	=	SYM
ejpam-619	68	6	0	0	NUM
ejpam-619	68	7	,	,	PUNCT
ejpam-619	68	8	(	(	PUNCT
ejpam-619	68	9	1	1	X
ejpam-619	68	10	)	)	PUNCT
ejpam-619	68	11	v̄t	v̄t	DET
ejpam-619	68	12	h	h	NOUN
ejpam-619	68	13	(	(	PUNCT
ejpam-619	68	14	x̄	x̄	PROPN
ejpam-619	68	15	)	)	PUNCT
ejpam-619	68	16	=	=	SYM
ejpam-619	68	17	0	0	NUM
ejpam-619	68	18	,	,	PUNCT
ejpam-619	68	19	(	(	PUNCT
ejpam-619	68	20	2	2	X
ejpam-619	68	21	)	)	PUNCT
ejpam-619	68	22	v̄	v̄	NOUN
ejpam-619	68	23	≧	≧	NOUN
ejpam-619	68	24	0	0	NUM
ejpam-619	68	25	,	,	PUNCT
ejpam-619	68	26	h	h	NOUN
ejpam-619	68	27	(	(	PUNCT
ejpam-619	68	28	x̄)≦	x̄)≦	X
ejpam-619	68	29	0	0	NUM
ejpam-619	68	30	,	,	PUNCT
ejpam-619	68	31	(	(	PUNCT
ejpam-619	68	32	3	3	X
ejpam-619	68	33	)	)	PUNCT
ejpam-619	68	34	where	where	SCONJ
ejpam-619	68	35	∇h	∇h	PROPN
ejpam-619	68	36	(	(	PUNCT
ejpam-619	68	37	x̄	x̄	PROPN
ejpam-619	68	38	)	)	PUNCT
ejpam-619	68	39	denotes	denote	VERB
ejpam-619	68	40	the	the	DET
ejpam-619	68	41	n×m	n×m	PROPN
ejpam-619	68	42	matrix	matrix	NOUN
ejpam-619	68	43	[	[	X
ejpam-619	68	44	∇h1	∇h1	X
ejpam-619	68	45	(	(	PUNCT
ejpam-619	68	46	x̄),∇h2	x̄),∇h2	PROPN
ejpam-619	68	47	(	(	PUNCT
ejpam-619	68	48	x̄	x̄	PROPN
ejpam-619	68	49	)	)	PUNCT
ejpam-619	68	50	,	,	PUNCT
ejpam-619	68	51	.	.	PUNCT
ejpam-619	68	52	.	.	PUNCT
ejpam-619	68	53	.	.	PUNCT
ejpam-619	69	1	,	,	PUNCT
ejpam-619	69	2	∇hm	∇hm	PROPN
ejpam-619	69	3	(	(	PUNCT
ejpam-619	69	4	x̄	x̄	PROPN
ejpam-619	69	5	)	)	PUNCT
ejpam-619	69	6	]	]	PUNCT
ejpam-619	69	7	.	.	PUNCT
ejpam-619	70	1	3	3	X
ejpam-619	70	2	.	.	X
ejpam-619	70	3	sufficient	sufficient	ADJ
ejpam-619	70	4	optimality	optimality	NOUN
ejpam-619	70	5	conditions	condition	NOUN
ejpam-619	70	6	in	in	ADP
ejpam-619	70	7	this	this	DET
ejpam-619	70	8	section	section	NOUN
ejpam-619	70	9	,	,	PUNCT
ejpam-619	70	10	we	we	PRON
ejpam-619	70	11	establish	establish	VERB
ejpam-619	70	12	kuhn	kuhn	PROPN
ejpam-619	70	13	-	-	PUNCT
ejpam-619	70	14	tucker	tucker	PROPN
ejpam-619	70	15	sufficient	sufficient	ADJ
ejpam-619	70	16	optimality	optimality	NOUN
ejpam-619	70	17	conditions	condition	NOUN
ejpam-619	70	18	for	for	ADP
ejpam-619	70	19	(	(	PUNCT
ejpam-619	70	20	np	np	INTJ
ejpam-619	70	21	)	)	PUNCT
ejpam-619	70	22	under	under	ADP
ejpam-619	70	23	(	(	PUNCT
ejpam-619	70	24	f	f	X
ejpam-619	70	25	,	,	PUNCT
ejpam-619	70	26	α	α	PROPN
ejpam-619	70	27	,	,	PUNCT
ejpam-619	70	28	β	β	PROPN
ejpam-619	70	29	,	,	PUNCT
ejpam-619	70	30	ρ	ρ	PROPN
ejpam-619	70	31	,	,	PUNCT
ejpam-619	70	32	d)-convexity	d)-convexity	NOUN
ejpam-619	70	33	assumptions	assumption	NOUN
ejpam-619	70	34	.	.	PUNCT
ejpam-619	71	1	theorem	theorem	NOUN
ejpam-619	71	2	1	1	NUM
ejpam-619	71	3	.	.	PUNCT
ejpam-619	72	1	let	let	VERB
ejpam-619	72	2	x̄	x̄	PRON
ejpam-619	72	3	∈	∈	PROPN
ejpam-619	72	4	s	s	PART
ejpam-619	72	5	and	and	CCONJ
ejpam-619	72	6	v̄	v̄	PROPN
ejpam-619	72	7	∈	∈	PROPN
ejpam-619	72	8	rm	rm	NOUN
ejpam-619	72	9	satisfy	satisfy	NOUN
ejpam-619	72	10	(	(	PUNCT
ejpam-619	72	11	1)-(3	1)-(3	NUM
ejpam-619	72	12	)	)	PUNCT
ejpam-619	72	13	.	.	PUNCT
ejpam-619	73	1	if	if	SCONJ
ejpam-619	73	2	(	(	PUNCT
ejpam-619	73	3	i	i	NOUN
ejpam-619	73	4	)	)	PUNCT
ejpam-619	73	5	φ	φ	PROPN
ejpam-619	73	6	is	be	AUX
ejpam-619	73	7	higher	high	ADJ
ejpam-619	73	8	-	-	PUNCT
ejpam-619	73	9	order	order	NOUN
ejpam-619	73	10	(	(	PUNCT
ejpam-619	73	11	f	f	X
ejpam-619	73	12	,	,	PUNCT
ejpam-619	73	13	α	α	PROPN
ejpam-619	73	14	,	,	PUNCT
ejpam-619	73	15	β	β	X
ejpam-619	73	16	,	,	PUNCT
ejpam-619	73	17	ρ1	ρ1	NOUN
ejpam-619	73	18	,	,	PUNCT
ejpam-619	73	19	d)-convex	d)-convex	PUNCT
ejpam-619	73	20	at	at	ADP
ejpam-619	73	21	x̄	x̄	NOUN
ejpam-619	73	22	with	with	ADP
ejpam-619	73	23	respect	respect	NOUN
ejpam-619	73	24	to	to	ADP
ejpam-619	73	25	k	k	PROPN
ejpam-619	73	26	,	,	PUNCT
ejpam-619	73	27	(	(	PUNCT
ejpam-619	73	28	ii	ii	NOUN
ejpam-619	73	29	)	)	PUNCT
ejpam-619	73	30	v̄t	v̄t	DET
ejpam-619	73	31	h	h	NOUN
ejpam-619	73	32	is	be	AUX
ejpam-619	73	33	higher	high	ADJ
ejpam-619	73	34	-	-	PUNCT
ejpam-619	73	35	order	order	NOUN
ejpam-619	73	36	(	(	PUNCT
ejpam-619	73	37	f	f	X
ejpam-619	73	38	,	,	PUNCT
ejpam-619	73	39	α	α	PROPN
ejpam-619	73	40	,	,	PUNCT
ejpam-619	73	41	β	β	X
ejpam-619	73	42	,	,	PUNCT
ejpam-619	73	43	ρ2	ρ2	NOUN
ejpam-619	73	44	,	,	PUNCT
ejpam-619	73	45	d)-convex	d)-convex	NOUN
ejpam-619	73	46	at	at	ADP
ejpam-619	73	47	x̄	x̄	NOUN
ejpam-619	73	48	with	with	ADP
ejpam-619	73	49	respect	respect	NOUN
ejpam-619	73	50	to	to	ADP
ejpam-619	73	51	−k	−k	ADJ
ejpam-619	73	52	,	,	PUNCT
ejpam-619	73	53	and	and	CCONJ
ejpam-619	73	54	(	(	PUNCT
ejpam-619	73	55	iii	iii	NOUN
ejpam-619	73	56	)	)	PUNCT
ejpam-619	73	57	ρ1	ρ1	NOUN
ejpam-619	73	58	+	+	NOUN
ejpam-619	73	59	ρ2	ρ2	NOUN
ejpam-619	73	60	≧	≧	NOUN
ejpam-619	73	61	0	0	NUM
ejpam-619	73	62	,	,	PUNCT
ejpam-619	73	63	then	then	ADV
ejpam-619	73	64	x̄	x̄	PROPN
ejpam-619	73	65	is	be	AUX
ejpam-619	73	66	an	an	DET
ejpam-619	73	67	optimal	optimal	ADJ
ejpam-619	73	68	solution	solution	NOUN
ejpam-619	73	69	of	of	ADP
ejpam-619	73	70	the	the	DET
ejpam-619	73	71	problem	problem	NOUN
ejpam-619	73	72	(	(	PUNCT
ejpam-619	73	73	np	np	INTJ
ejpam-619	73	74	)	)	PUNCT
ejpam-619	73	75	.	.	PUNCT
ejpam-619	74	1	proof	proof	NOUN
ejpam-619	74	2	.	.	PUNCT
ejpam-619	75	1	let	let	VERB
ejpam-619	75	2	x̄	x̄	PRON
ejpam-619	75	3	∈	∈	PROPN
ejpam-619	75	4	s.	s.	PROPN
ejpam-619	75	5	since	since	SCONJ
ejpam-619	75	6	φ	φ	PROPN
ejpam-619	75	7	is	be	AUX
ejpam-619	75	8	higher	high	ADJ
ejpam-619	75	9	-	-	PUNCT
ejpam-619	75	10	order	order	NOUN
ejpam-619	75	11	(	(	PUNCT
ejpam-619	75	12	f	f	X
ejpam-619	75	13	,	,	PUNCT
ejpam-619	75	14	α	α	PROPN
ejpam-619	75	15	,	,	PUNCT
ejpam-619	75	16	β	β	X
ejpam-619	75	17	,	,	PUNCT
ejpam-619	75	18	ρ1	ρ1	NOUN
ejpam-619	75	19	,	,	PUNCT
ejpam-619	75	20	d)-convex	d)-convex	PUNCT
ejpam-619	75	21	at	at	ADP
ejpam-619	75	22	x̄	x̄	NOUN
ejpam-619	75	23	with	with	ADP
ejpam-619	75	24	respect	respect	NOUN
ejpam-619	75	25	to	to	ADP
ejpam-619	75	26	k	k	PROPN
ejpam-619	75	27	,	,	PUNCT
ejpam-619	75	28	for	for	ADP
ejpam-619	75	29	all	all	DET
ejpam-619	75	30	x	x	SYM
ejpam-619	75	31	∈	∈	PROPN
ejpam-619	75	32	s	s	X
ejpam-619	75	33	,	,	PUNCT
ejpam-619	75	34	we	we	PRON
ejpam-619	75	35	have	have	VERB
ejpam-619	75	36	φ(x)−φ	φ(x)−φ	PROPN
ejpam-619	75	37	(	(	PUNCT
ejpam-619	75	38	x̄	x̄	PROPN
ejpam-619	75	39	)	)	PUNCT
ejpam-619	75	40	≧	≧	X
ejpam-619	76	1	f	f	PROPN
ejpam-619	76	2	�	�	PROPN
ejpam-619	76	3	x	x	PROPN
ejpam-619	76	4	,	,	PUNCT
ejpam-619	76	5	x̄;α(x	x̄;α(x	PROPN
ejpam-619	76	6	,	,	PUNCT
ejpam-619	76	7	x̄)[∇φ	x̄)[∇φ	PROPN
ejpam-619	76	8	(	(	PUNCT
ejpam-619	76	9	x̄	x̄	PROPN
ejpam-619	76	10	)	)	PUNCT
ejpam-619	77	1	+	+	X
ejpam-619	77	2	∇pk	∇pk	NOUN
ejpam-619	77	3	(	(	PUNCT
ejpam-619	77	4	x̄	x̄	NOUN
ejpam-619	77	5	,	,	PUNCT
ejpam-619	77	6	p	p	NOUN
ejpam-619	77	7	)	)	PUNCT
ejpam-619	77	8	]	]	PUNCT
ejpam-619	77	9	�	�	PROPN
ejpam-619	77	10	+	+	CCONJ
ejpam-619	77	11	β(x	β(x	NOUN
ejpam-619	77	12	,	,	PUNCT
ejpam-619	77	13	x̄	x̄	NUM
ejpam-619	77	14	)	)	PUNCT
ejpam-619	77	15	�	�	PROPN
ejpam-619	78	1	k	k	PROPN
ejpam-619	78	2	(	(	PUNCT
ejpam-619	78	3	x̄	x̄	NOUN
ejpam-619	78	4	,	,	PUNCT
ejpam-619	78	5	p)−	p)−	PROPN
ejpam-619	78	6	pt∇pk	pt∇pk	PROPN
ejpam-619	78	7	(	(	PUNCT
ejpam-619	78	8	x̄	x̄	NOUN
ejpam-619	78	9	,	,	PUNCT
ejpam-619	78	10	p	p	X
ejpam-619	78	11	)	)	PUNCT
ejpam-619	78	12	�	�	PROPN
ejpam-619	79	1	+	+	NOUN
ejpam-619	79	2	ρ1d2(x	ρ1d2(x	PROPN
ejpam-619	79	3	,	,	PUNCT
ejpam-619	79	4	x̄	x̄	PROPN
ejpam-619	79	5	)	)	PUNCT
ejpam-619	79	6	.	.	PUNCT
ejpam-619	80	1	(	(	PUNCT
ejpam-619	80	2	4	4	X
ejpam-619	80	3	)	)	PUNCT
ejpam-619	80	4	t.	t.	NOUN
ejpam-619	80	5	gulati	gulati	PROPN
ejpam-619	80	6	,	,	PUNCT
ejpam-619	80	7	h.	h.	PROPN
ejpam-619	80	8	saini	saini	PROPN
ejpam-619	80	9	/	/	SYM
ejpam-619	80	10	eur	eur	PROPN
ejpam-619	80	11	.	.	PUNCT
ejpam-619	81	1	j.	j.	PROPN
ejpam-619	81	2	pure	pure	PROPN
ejpam-619	81	3	appl	appl	PROPN
ejpam-619	81	4	.	.	PROPN
ejpam-619	81	5	math	math	PROPN
ejpam-619	81	6	,	,	PUNCT
ejpam-619	81	7	4	4	NUM
ejpam-619	81	8	(	(	PUNCT
ejpam-619	81	9	2011	2011	NUM
ejpam-619	81	10	)	)	PUNCT
ejpam-619	81	11	,	,	PUNCT
ejpam-619	81	12	266	266	NUM
ejpam-619	81	13	-	-	SYM
ejpam-619	81	14	275	275	NUM
ejpam-619	81	15	269	269	NUM
ejpam-619	81	16	using	use	VERB
ejpam-619	81	17	(	(	PUNCT
ejpam-619	81	18	1	1	NUM
ejpam-619	81	19	)	)	PUNCT
ejpam-619	81	20	,	,	PUNCT
ejpam-619	81	21	we	we	PRON
ejpam-619	81	22	get	get	VERB
ejpam-619	81	23	φ(x)−φ	φ(x)−φ	PROPN
ejpam-619	81	24	(	(	PUNCT
ejpam-619	81	25	x̄	x̄	PROPN
ejpam-619	81	26	)	)	PUNCT
ejpam-619	81	27	≧	≧	X
ejpam-619	82	1	f	f	PROPN
ejpam-619	82	2	�	�	PROPN
ejpam-619	82	3	x	x	PROPN
ejpam-619	82	4	,	,	PUNCT
ejpam-619	82	5	x̄;α(x	x̄;α(x	PROPN
ejpam-619	82	6	,	,	PUNCT
ejpam-619	82	7	x̄)[−∇h	x̄)[−∇h	PROPN
ejpam-619	82	8	(	(	PUNCT
ejpam-619	82	9	x̄)v̄+∇pk	x̄)v̄+∇pk	PROPN
ejpam-619	82	10	(	(	PUNCT
ejpam-619	82	11	x̄	x̄	NOUN
ejpam-619	82	12	,	,	PUNCT
ejpam-619	82	13	p	p	X
ejpam-619	82	14	)	)	PUNCT
ejpam-619	82	15	]	]	PUNCT
ejpam-619	82	16	�	�	PROPN
ejpam-619	82	17	+	+	CCONJ
ejpam-619	82	18	β(x	β(x	NOUN
ejpam-619	82	19	,	,	PUNCT
ejpam-619	82	20	x̄	x̄	NUM
ejpam-619	82	21	)	)	PUNCT
ejpam-619	82	22	�	�	PROPN
ejpam-619	83	1	k	k	PROPN
ejpam-619	83	2	(	(	PUNCT
ejpam-619	83	3	x̄	x̄	NOUN
ejpam-619	83	4	,	,	PUNCT
ejpam-619	83	5	p)−	p)−	PROPN
ejpam-619	83	6	pt∇pk	pt∇pk	PROPN
ejpam-619	83	7	(	(	PUNCT
ejpam-619	83	8	x̄	x̄	NOUN
ejpam-619	83	9	,	,	PUNCT
ejpam-619	83	10	p	p	X
ejpam-619	83	11	)	)	PUNCT
ejpam-619	83	12	�	�	PROPN
ejpam-619	84	1	+	+	NOUN
ejpam-619	84	2	ρ1d2(x	ρ1d2(x	PROPN
ejpam-619	84	3	,	,	PUNCT
ejpam-619	84	4	x̄	x̄	PROPN
ejpam-619	84	5	)	)	PUNCT
ejpam-619	84	6	.	.	PUNCT
ejpam-619	85	1	(	(	PUNCT
ejpam-619	85	2	5	5	NUM
ejpam-619	85	3	)	)	PUNCT
ejpam-619	85	4	also	also	ADV
ejpam-619	85	5	,	,	PUNCT
ejpam-619	85	6	v̄t	v̄t	DET
ejpam-619	85	7	h	h	NOUN
ejpam-619	85	8	is	be	AUX
ejpam-619	85	9	higher	high	ADJ
ejpam-619	85	10	-	-	PUNCT
ejpam-619	85	11	order	order	NOUN
ejpam-619	85	12	(	(	PUNCT
ejpam-619	85	13	f	f	X
ejpam-619	85	14	,	,	PUNCT
ejpam-619	85	15	α	α	PROPN
ejpam-619	85	16	,	,	PUNCT
ejpam-619	85	17	β	β	X
ejpam-619	85	18	,	,	PUNCT
ejpam-619	85	19	ρ2	ρ2	NOUN
ejpam-619	85	20	,	,	PUNCT
ejpam-619	85	21	d)-convex	d)-convex	NOUN
ejpam-619	85	22	at	at	ADP
ejpam-619	85	23	x̄	x̄	NOUN
ejpam-619	85	24	with	with	ADP
ejpam-619	85	25	respect	respect	NOUN
ejpam-619	85	26	to	to	ADP
ejpam-619	85	27	−k	−k	NOUN
ejpam-619	85	28	.	.	PUNCT
ejpam-619	86	1	therefore	therefore	ADV
ejpam-619	86	2	v̄t	v̄t	DET
ejpam-619	86	3	h(x)−	h(x)−	PROPN
ejpam-619	86	4	v̄t	v̄t	DET
ejpam-619	86	5	h	h	PROPN
ejpam-619	86	6	(	(	PUNCT
ejpam-619	86	7	x̄	x̄	PROPN
ejpam-619	86	8	)	)	PUNCT
ejpam-619	86	9	≧	≧	X
ejpam-619	87	1	f	f	PROPN
ejpam-619	87	2	�	�	PROPN
ejpam-619	87	3	x	x	PROPN
ejpam-619	87	4	,	,	PUNCT
ejpam-619	87	5	x̄;α(x	x̄;α(x	PROPN
ejpam-619	87	6	,	,	PUNCT
ejpam-619	87	7	x̄)[∇v̄t	x̄)[∇v̄t	PROPN
ejpam-619	87	8	h	h	PROPN
ejpam-619	87	9	(	(	PUNCT
ejpam-619	87	10	x̄)−∇pk	x̄)−∇pk	PROPN
ejpam-619	87	11	(	(	PUNCT
ejpam-619	87	12	x̄	x̄	NOUN
ejpam-619	87	13	,	,	PUNCT
ejpam-619	87	14	p	p	X
ejpam-619	87	15	)	)	PUNCT
ejpam-619	87	16	]	]	PUNCT
ejpam-619	87	17	�	�	PROPN
ejpam-619	87	18	−	−	PROPN
ejpam-619	87	19	β(x	β(x	PROPN
ejpam-619	87	20	,	,	PUNCT
ejpam-619	87	21	x̄	x̄	NUM
ejpam-619	87	22	)	)	PUNCT
ejpam-619	87	23	�	�	PROPN
ejpam-619	88	1	k	k	PROPN
ejpam-619	88	2	(	(	PUNCT
ejpam-619	88	3	x̄	x̄	NOUN
ejpam-619	88	4	,	,	PUNCT
ejpam-619	88	5	p)−	p)−	PROPN
ejpam-619	88	6	pt∇pk	pt∇pk	PROPN
ejpam-619	88	7	(	(	PUNCT
ejpam-619	88	8	x̄	x̄	NOUN
ejpam-619	88	9	,	,	PUNCT
ejpam-619	88	10	p	p	X
ejpam-619	88	11	)	)	PUNCT
ejpam-619	88	12	�	�	PROPN
ejpam-619	89	1	+	+	NOUN
ejpam-619	89	2	ρ2d2(x	ρ2d2(x	X
ejpam-619	89	3	,	,	PUNCT
ejpam-619	89	4	x̄	x̄	PROPN
ejpam-619	89	5	)	)	PUNCT
ejpam-619	89	6	.	.	PUNCT
ejpam-619	90	1	(	(	PUNCT
ejpam-619	90	2	6	6	NUM
ejpam-619	90	3	)	)	PUNCT
ejpam-619	90	4	since	since	SCONJ
ejpam-619	90	5	v̄t	v̄t	DET
ejpam-619	90	6	h	h	PROPN
ejpam-619	90	7	(	(	PUNCT
ejpam-619	90	8	x̄	x̄	PROPN
ejpam-619	90	9	)	)	PUNCT
ejpam-619	90	10	=	=	SYM
ejpam-619	90	11	0	0	NUM
ejpam-619	90	12	,	,	PUNCT
ejpam-619	90	13	v̄	v̄	NOUN
ejpam-619	90	14	≧	≧	NOUN
ejpam-619	90	15	0	0	NUM
ejpam-619	90	16	and	and	CCONJ
ejpam-619	90	17	h(x)≦	h(x)≦	X
ejpam-619	90	18	0	0	NUM
ejpam-619	90	19	,	,	PUNCT
ejpam-619	90	20	we	we	PRON
ejpam-619	90	21	get	get	VERB
ejpam-619	90	22	0	0	NUM
ejpam-619	90	23	≧	≧	NUM
ejpam-619	91	1	f	f	PROPN
ejpam-619	91	2	�	�	PROPN
ejpam-619	91	3	x	x	PROPN
ejpam-619	91	4	,	,	PUNCT
ejpam-619	91	5	x̄;α(x	x̄;α(x	PROPN
ejpam-619	91	6	,	,	PUNCT
ejpam-619	91	7	x̄)[∇v̄t	x̄)[∇v̄t	PROPN
ejpam-619	91	8	h	h	PROPN
ejpam-619	91	9	(	(	PUNCT
ejpam-619	91	10	x̄)−∇pk	x̄)−∇pk	PROPN
ejpam-619	91	11	(	(	PUNCT
ejpam-619	91	12	x̄	x̄	NOUN
ejpam-619	91	13	,	,	PUNCT
ejpam-619	91	14	p	p	X
ejpam-619	91	15	)	)	PUNCT
ejpam-619	91	16	]	]	PUNCT
ejpam-619	91	17	�	�	PROPN
ejpam-619	91	18	−	−	PROPN
ejpam-619	91	19	β(x	β(x	PROPN
ejpam-619	91	20	,	,	PUNCT
ejpam-619	91	21	x̄	x̄	NUM
ejpam-619	91	22	)	)	PUNCT
ejpam-619	91	23	�	�	PROPN
ejpam-619	92	1	k	k	PROPN
ejpam-619	92	2	(	(	PUNCT
ejpam-619	92	3	x̄	x̄	NOUN
ejpam-619	92	4	,	,	PUNCT
ejpam-619	92	5	p)−	p)−	PROPN
ejpam-619	92	6	pt∇pk	pt∇pk	PROPN
ejpam-619	92	7	(	(	PUNCT
ejpam-619	92	8	x̄	x̄	NOUN
ejpam-619	92	9	,	,	PUNCT
ejpam-619	92	10	p	p	X
ejpam-619	92	11	)	)	PUNCT
ejpam-619	92	12	�	�	PROPN
ejpam-619	93	1	+	+	NOUN
ejpam-619	93	2	ρ2d2(x	ρ2d2(x	X
ejpam-619	93	3	,	,	PUNCT
ejpam-619	93	4	x̄	x̄	PROPN
ejpam-619	93	5	)	)	PUNCT
ejpam-619	93	6	.	.	PUNCT
ejpam-619	94	1	(	(	PUNCT
ejpam-619	94	2	7	7	X
ejpam-619	94	3	)	)	PUNCT
ejpam-619	94	4	adding	add	VERB
ejpam-619	94	5	the	the	DET
ejpam-619	94	6	inequalities	inequality	NOUN
ejpam-619	94	7	(	(	PUNCT
ejpam-619	94	8	5	5	NUM
ejpam-619	94	9	)	)	PUNCT
ejpam-619	94	10	and	and	CCONJ
ejpam-619	94	11	(	(	PUNCT
ejpam-619	94	12	7	7	NUM
ejpam-619	94	13	)	)	PUNCT
ejpam-619	94	14	,	,	PUNCT
ejpam-619	94	15	we	we	PRON
ejpam-619	94	16	obtain	obtain	VERB
ejpam-619	94	17	φ(x)−φ	φ(x)−φ	PROPN
ejpam-619	94	18	(	(	PUNCT
ejpam-619	94	19	x̄)≧	x̄)≧	PROPN
ejpam-619	94	20	(	(	PUNCT
ejpam-619	94	21	ρ1+ρ2)d	ρ1+ρ2)d	PROPN
ejpam-619	94	22	2(x	2(x	NUM
ejpam-619	94	23	,	,	PUNCT
ejpam-619	94	24	x̄	x̄	PROPN
ejpam-619	94	25	)	)	PUNCT
ejpam-619	94	26	,	,	PUNCT
ejpam-619	94	27	which	which	PRON
ejpam-619	94	28	by	by	ADP
ejpam-619	94	29	hypothesis	hypothesis	NOUN
ejpam-619	94	30	(	(	PUNCT
ejpam-619	94	31	iii	iii	NOUN
ejpam-619	94	32	)	)	PUNCT
ejpam-619	94	33	implies	imply	VERB
ejpam-619	94	34	,	,	PUNCT
ejpam-619	94	35	φ(x)≧	φ(x)≧	PROPN
ejpam-619	94	36	φ	φ	PROPN
ejpam-619	94	37	(	(	PUNCT
ejpam-619	94	38	x̄	x̄	PROPN
ejpam-619	94	39	)	)	PUNCT
ejpam-619	94	40	.	.	PUNCT
ejpam-619	95	1	hence	hence	ADV
ejpam-619	95	2	x̄	x̄	PROPN
ejpam-619	95	3	is	be	AUX
ejpam-619	95	4	an	an	DET
ejpam-619	95	5	optimal	optimal	ADJ
ejpam-619	95	6	solution	solution	NOUN
ejpam-619	95	7	of	of	ADP
ejpam-619	95	8	the	the	DET
ejpam-619	95	9	problem	problem	NOUN
ejpam-619	95	10	(	(	PUNCT
ejpam-619	95	11	np	np	INTJ
ejpam-619	95	12	)	)	PUNCT
ejpam-619	95	13	.	.	PUNCT
ejpam-619	96	1	4	4	X
ejpam-619	96	2	.	.	X
ejpam-619	96	3	mond	mond	PROPN
ejpam-619	96	4	weir	weir	PROPN
ejpam-619	96	5	duality	duality	PROPN
ejpam-619	96	6	in	in	ADP
ejpam-619	96	7	this	this	DET
ejpam-619	96	8	section	section	NOUN
ejpam-619	96	9	,	,	PUNCT
ejpam-619	96	10	we	we	PRON
ejpam-619	96	11	establish	establish	VERB
ejpam-619	96	12	weak	weak	ADJ
ejpam-619	96	13	and	and	CCONJ
ejpam-619	96	14	strong	strong	ADJ
ejpam-619	96	15	duality	duality	NOUN
ejpam-619	96	16	theorems	theorem	NOUN
ejpam-619	96	17	for	for	ADP
ejpam-619	96	18	the	the	DET
ejpam-619	96	19	following	follow	VERB
ejpam-619	96	20	mond	mond	PROPN
ejpam-619	96	21	weir	weir	PROPN
ejpam-619	96	22	dual	dual	PROPN
ejpam-619	96	23	(	(	PUNCT
ejpam-619	96	24	md	md	PROPN
ejpam-619	96	25	)	)	PUNCT
ejpam-619	96	26	for	for	ADP
ejpam-619	96	27	(	(	PUNCT
ejpam-619	96	28	np	np	INTJ
ejpam-619	96	29	):	):	PUNCT
ejpam-619	96	30	(	(	PUNCT
ejpam-619	96	31	md	md	PROPN
ejpam-619	96	32	)	)	PUNCT
ejpam-619	96	33	maximize	maximize	VERB
ejpam-619	96	34	φ(u	φ(u	NOUN
ejpam-619	96	35	)	)	PUNCT
ejpam-619	96	36	,	,	PUNCT
ejpam-619	96	37	subject	subject	ADJ
ejpam-619	96	38	to	to	ADP
ejpam-619	96	39	∇φ(u	∇φ(u	NOUN
ejpam-619	96	40	)	)	PUNCT
ejpam-619	97	1	+	+	NOUN
ejpam-619	97	2	∇h(u)v	∇h(u)v	NOUN
ejpam-619	97	3	=	=	SYM
ejpam-619	97	4	0	0	PROPN
ejpam-619	97	5	,	,	PUNCT
ejpam-619	97	6	(	(	PUNCT
ejpam-619	97	7	8)	8)	NUM
ejpam-619	97	8	vt	vt	PROPN
ejpam-619	97	9	h(u)≧	h(u)≧	ADP
ejpam-619	97	10	0	0	NUM
ejpam-619	97	11	,	,	PUNCT
ejpam-619	97	12	(	(	PUNCT
ejpam-619	97	13	9	9	X
ejpam-619	97	14	)	)	PUNCT
ejpam-619	97	15	u	u	NOUN
ejpam-619	97	16	∈	∈	PROPN
ejpam-619	97	17	x	x	X
ejpam-619	97	18	,	,	PUNCT
ejpam-619	97	19	v	v	PART
ejpam-619	97	20	≧	≧	NOUN
ejpam-619	97	21	0	0	NUM
ejpam-619	97	22	,	,	PUNCT
ejpam-619	97	23	v	v	PROPN
ejpam-619	97	24	∈	∈	PROPN
ejpam-619	97	25	rm	rm	NOUN
ejpam-619	97	26	.	.	PUNCT
ejpam-619	98	1	(	(	PUNCT
ejpam-619	98	2	10	10	NUM
ejpam-619	98	3	)	)	PUNCT
ejpam-619	98	4	theorem	theorem	NOUN
ejpam-619	98	5	2	2	NUM
ejpam-619	98	6	(	(	PUNCT
ejpam-619	98	7	weak	weak	ADJ
ejpam-619	98	8	duality	duality	NOUN
ejpam-619	98	9	)	)	PUNCT
ejpam-619	98	10	.	.	PUNCT
ejpam-619	99	1	let	let	VERB
ejpam-619	99	2	x	x	PRON
ejpam-619	99	3	and	and	CCONJ
ejpam-619	99	4	(	(	PUNCT
ejpam-619	99	5	u	u	NOUN
ejpam-619	99	6	,	,	PUNCT
ejpam-619	99	7	v	v	NOUN
ejpam-619	99	8	)	)	PUNCT
ejpam-619	99	9	be	be	AUX
ejpam-619	99	10	feasible	feasible	ADJ
ejpam-619	99	11	solutions	solution	NOUN
ejpam-619	99	12	of	of	ADP
ejpam-619	99	13	(	(	PUNCT
ejpam-619	99	14	np	np	INTJ
ejpam-619	99	15	)	)	PUNCT
ejpam-619	99	16	and	and	CCONJ
ejpam-619	99	17	(	(	PUNCT
ejpam-619	99	18	md	md	PROPN
ejpam-619	99	19	)	)	PUNCT
ejpam-619	99	20	respectively	respectively	ADV
ejpam-619	99	21	.	.	PUNCT
ejpam-619	100	1	let	let	VERB
ejpam-619	100	2	(	(	PUNCT
ejpam-619	100	3	i	i	NOUN
ejpam-619	100	4	)	)	PUNCT
ejpam-619	100	5	φ	φ	PROPN
ejpam-619	100	6	be	be	VERB
ejpam-619	100	7	higher	high	ADJ
ejpam-619	100	8	-	-	PUNCT
ejpam-619	100	9	order	order	NOUN
ejpam-619	100	10	(	(	PUNCT
ejpam-619	100	11	f	f	X
ejpam-619	100	12	,	,	PUNCT
ejpam-619	100	13	α	α	PROPN
ejpam-619	100	14	,	,	PUNCT
ejpam-619	100	15	β	β	X
ejpam-619	100	16	,	,	PUNCT
ejpam-619	100	17	ρ1	ρ1	NOUN
ejpam-619	100	18	,	,	PUNCT
ejpam-619	100	19	d)-convex	d)-convex	PUNCT
ejpam-619	100	20	at	at	ADP
ejpam-619	100	21	u	u	NOUN
ejpam-619	100	22	with	with	ADP
ejpam-619	100	23	respect	respect	NOUN
ejpam-619	100	24	to	to	ADP
ejpam-619	100	25	k	k	PROPN
ejpam-619	100	26	,	,	PUNCT
ejpam-619	100	27	(	(	PUNCT
ejpam-619	100	28	ii	ii	NOUN
ejpam-619	100	29	)	)	PUNCT
ejpam-619	100	30	vt	vt	PROPN
ejpam-619	100	31	h	h	PROPN
ejpam-619	100	32	be	be	AUX
ejpam-619	100	33	higher	high	ADJ
ejpam-619	100	34	-	-	PUNCT
ejpam-619	100	35	order	order	NOUN
ejpam-619	100	36	(	(	PUNCT
ejpam-619	100	37	f	f	X
ejpam-619	100	38	,	,	PUNCT
ejpam-619	100	39	α	α	PROPN
ejpam-619	100	40	,	,	PUNCT
ejpam-619	100	41	β	β	X
ejpam-619	100	42	,	,	PUNCT
ejpam-619	100	43	ρ2	ρ2	NOUN
ejpam-619	100	44	,	,	PUNCT
ejpam-619	100	45	d)-convex	d)-convex	NOUN
ejpam-619	100	46	at	at	ADP
ejpam-619	100	47	u	u	NOUN
ejpam-619	100	48	with	with	ADP
ejpam-619	100	49	respect	respect	NOUN
ejpam-619	100	50	to	to	ADP
ejpam-619	100	51	−k	−k	ADJ
ejpam-619	100	52	,	,	PUNCT
ejpam-619	100	53	and	and	CCONJ
ejpam-619	100	54	(	(	PUNCT
ejpam-619	100	55	iii	iii	NOUN
ejpam-619	100	56	)	)	PUNCT
ejpam-619	100	57	ρ1	ρ1	NOUN
ejpam-619	100	58	+	+	NOUN
ejpam-619	100	59	ρ2	ρ2	NOUN
ejpam-619	100	60	≧	≧	NOUN
ejpam-619	100	61	0	0	X
ejpam-619	100	62	.	.	PUNCT
ejpam-619	101	1	then	then	ADV
ejpam-619	101	2	φ(x)≧	φ(x)≧	PROPN
ejpam-619	101	3	φ(u	φ(u	NOUN
ejpam-619	101	4	)	)	PUNCT
ejpam-619	101	5	.	.	PUNCT
ejpam-619	102	1	t.	t.	PROPN
ejpam-619	102	2	gulati	gulati	PROPN
ejpam-619	102	3	,	,	PUNCT
ejpam-619	102	4	h.	h.	PROPN
ejpam-619	102	5	saini	saini	PROPN
ejpam-619	102	6	/	/	SYM
ejpam-619	102	7	eur	eur	PROPN
ejpam-619	102	8	.	.	PUNCT
ejpam-619	103	1	j.	j.	PROPN
ejpam-619	103	2	pure	pure	PROPN
ejpam-619	103	3	appl	appl	PROPN
ejpam-619	103	4	.	.	PROPN
ejpam-619	103	5	math	math	PROPN
ejpam-619	103	6	,	,	PUNCT
ejpam-619	103	7	4	4	NUM
ejpam-619	103	8	(	(	PUNCT
ejpam-619	103	9	2011	2011	NUM
ejpam-619	103	10	)	)	PUNCT
ejpam-619	103	11	,	,	PUNCT
ejpam-619	103	12	266	266	NUM
ejpam-619	103	13	-	-	SYM
ejpam-619	103	14	275	275	NUM
ejpam-619	103	15	270	270	NUM
ejpam-619	103	16	proof	proof	NOUN
ejpam-619	103	17	.	.	PUNCT
ejpam-619	104	1	by	by	ADP
ejpam-619	104	2	hypothesis	hypothesis	NOUN
ejpam-619	104	3	(	(	PUNCT
ejpam-619	104	4	i	i	NOUN
ejpam-619	104	5	)	)	PUNCT
ejpam-619	104	6	,	,	PUNCT
ejpam-619	104	7	we	we	PRON
ejpam-619	104	8	have	have	VERB
ejpam-619	104	9	φ(x)−φ(u	φ(x)−φ(u	NOUN
ejpam-619	104	10	)	)	PUNCT
ejpam-619	104	11	≧	≧	X
ejpam-619	105	1	f	f	PROPN
ejpam-619	105	2	�	�	PROPN
ejpam-619	105	3	x	x	PROPN
ejpam-619	105	4	,	,	PUNCT
ejpam-619	105	5	u;α(x	u;α(x	PROPN
ejpam-619	105	6	,	,	PUNCT
ejpam-619	105	7	u)[∇φ(u	u)[∇φ(u	NOUN
ejpam-619	105	8	)	)	PUNCT
ejpam-619	106	1	+	+	VERB
ejpam-619	106	2	∇pk(u	∇pk(u	NOUN
ejpam-619	106	3	,	,	PUNCT
ejpam-619	106	4	p	p	NOUN
ejpam-619	106	5	)	)	PUNCT
ejpam-619	106	6	]	]	PUNCT
ejpam-619	106	7	�	�	PROPN
ejpam-619	106	8	+	+	CCONJ
ejpam-619	106	9	β(x	β(x	NOUN
ejpam-619	106	10	,	,	PUNCT
ejpam-619	106	11	u	u	NOUN
ejpam-619	106	12	)	)	PUNCT
ejpam-619	106	13	�	�	PROPN
ejpam-619	106	14	k(u	k(u	X
ejpam-619	106	15	,	,	PUNCT
ejpam-619	106	16	p)−	p)−	PROPN
ejpam-619	106	17	pt∇pk(u	pt∇pk(u	PROPN
ejpam-619	106	18	,	,	PUNCT
ejpam-619	106	19	p	p	X
ejpam-619	106	20	)	)	PUNCT
ejpam-619	106	21	�	�	PROPN
ejpam-619	107	1	+	+	NOUN
ejpam-619	107	2	ρ1d2(x	ρ1d2(x	PROPN
ejpam-619	107	3	,	,	PUNCT
ejpam-619	107	4	u	u	NOUN
ejpam-619	107	5	)	)	PUNCT
ejpam-619	107	6	.	.	PUNCT
ejpam-619	108	1	(	(	PUNCT
ejpam-619	108	2	11	11	NUM
ejpam-619	108	3	)	)	PUNCT
ejpam-619	108	4	also	also	ADV
ejpam-619	108	5	hypothesis	hypothesis	NOUN
ejpam-619	108	6	(	(	PUNCT
ejpam-619	108	7	ii	ii	NOUN
ejpam-619	108	8	)	)	PUNCT
ejpam-619	108	9	yields	yield	NOUN
ejpam-619	108	10	vt	vt	PROPN
ejpam-619	108	11	h(x)−	h(x)−	PROPN
ejpam-619	108	12	vt	vt	PROPN
ejpam-619	108	13	h(u	h(u	PROPN
ejpam-619	108	14	)	)	PUNCT
ejpam-619	108	15	≧	≧	X
ejpam-619	109	1	f	f	PROPN
ejpam-619	109	2	�	�	PROPN
ejpam-619	109	3	x	x	PROPN
ejpam-619	109	4	,	,	PUNCT
ejpam-619	109	5	u;α(x	u;α(x	PROPN
ejpam-619	109	6	,	,	PUNCT
ejpam-619	109	7	u)[∇vth(u)−∇pk(u	u)[∇vth(u)−∇pk(u	NOUN
ejpam-619	109	8	,	,	PUNCT
ejpam-619	109	9	p	p	NOUN
ejpam-619	109	10	)	)	PUNCT
ejpam-619	109	11	]	]	PUNCT
ejpam-619	109	12	�	�	PROPN
ejpam-619	109	13	−	−	PROPN
ejpam-619	109	14	β(x	β(x	PROPN
ejpam-619	109	15	,	,	PUNCT
ejpam-619	109	16	u	u	NOUN
ejpam-619	109	17	)	)	PUNCT
ejpam-619	109	18	�	�	PROPN
ejpam-619	109	19	k(u	k(u	X
ejpam-619	109	20	,	,	PUNCT
ejpam-619	109	21	p)−	p)−	PROPN
ejpam-619	109	22	pt∇pk(u	pt∇pk(u	PROPN
ejpam-619	109	23	,	,	PUNCT
ejpam-619	109	24	p	p	X
ejpam-619	109	25	)	)	PUNCT
ejpam-619	109	26	�	�	PROPN
ejpam-619	110	1	+	+	NOUN
ejpam-619	110	2	ρ2d2(x	ρ2d2(x	X
ejpam-619	110	3	,	,	PUNCT
ejpam-619	110	4	u	u	NOUN
ejpam-619	110	5	)	)	PUNCT
ejpam-619	110	6	.	.	PUNCT
ejpam-619	111	1	by	by	ADP
ejpam-619	111	2	(	(	PUNCT
ejpam-619	111	3	9	9	NUM
ejpam-619	111	4	)	)	PUNCT
ejpam-619	111	5	,	,	PUNCT
ejpam-619	111	6	(	(	PUNCT
ejpam-619	111	7	10	10	NUM
ejpam-619	111	8	)	)	PUNCT
ejpam-619	111	9	and	and	CCONJ
ejpam-619	111	10	h(x)≦	h(x)≦	X
ejpam-619	111	11	0	0	NUM
ejpam-619	111	12	,	,	PUNCT
ejpam-619	111	13	it	it	PRON
ejpam-619	111	14	follows	follow	VERB
ejpam-619	111	15	that	that	SCONJ
ejpam-619	111	16	0	0	NUM
ejpam-619	111	17	≧	≧	NUM
ejpam-619	112	1	f	f	PROPN
ejpam-619	112	2	�	�	PROPN
ejpam-619	112	3	x	x	PROPN
ejpam-619	112	4	,	,	PUNCT
ejpam-619	112	5	u;α(x	u;α(x	PROPN
ejpam-619	112	6	,	,	PUNCT
ejpam-619	112	7	u)[∇vt	u)[∇vt	NOUN
ejpam-619	112	8	h(u)−∇pk(u	h(u)−∇pk(u	PRON
ejpam-619	112	9	,	,	PUNCT
ejpam-619	112	10	p	p	X
ejpam-619	112	11	)	)	PUNCT
ejpam-619	112	12	]	]	PUNCT
ejpam-619	112	13	�	�	PROPN
ejpam-619	112	14	−	−	PROPN
ejpam-619	112	15	β(x	β(x	PROPN
ejpam-619	112	16	,	,	PUNCT
ejpam-619	112	17	u	u	NOUN
ejpam-619	112	18	)	)	PUNCT
ejpam-619	112	19	�	�	PROPN
ejpam-619	112	20	k(u	k(u	X
ejpam-619	112	21	,	,	PUNCT
ejpam-619	112	22	p)−	p)−	PROPN
ejpam-619	112	23	pt∇pk(u	pt∇pk(u	PROPN
ejpam-619	112	24	,	,	PUNCT
ejpam-619	112	25	p	p	X
ejpam-619	112	26	)	)	PUNCT
ejpam-619	112	27	�	�	PROPN
ejpam-619	113	1	+	+	NOUN
ejpam-619	113	2	ρ2d2(x	ρ2d2(x	X
ejpam-619	113	3	,	,	PUNCT
ejpam-619	113	4	u	u	NOUN
ejpam-619	113	5	)	)	PUNCT
ejpam-619	113	6	.	.	PUNCT
ejpam-619	114	1	(	(	PUNCT
ejpam-619	114	2	12	12	NUM
ejpam-619	114	3	)	)	PUNCT
ejpam-619	114	4	adding	add	VERB
ejpam-619	114	5	the	the	DET
ejpam-619	114	6	inequalities	inequality	NOUN
ejpam-619	114	7	(	(	PUNCT
ejpam-619	114	8	11	11	NUM
ejpam-619	114	9	)	)	PUNCT
ejpam-619	114	10	,	,	PUNCT
ejpam-619	114	11	(	(	PUNCT
ejpam-619	114	12	12	12	NUM
ejpam-619	114	13	)	)	PUNCT
ejpam-619	114	14	and	and	CCONJ
ejpam-619	114	15	applying	apply	VERB
ejpam-619	114	16	the	the	DET
ejpam-619	114	17	properties	property	NOUN
ejpam-619	114	18	of	of	ADP
ejpam-619	114	19	sublinear	sublinear	NOUN
ejpam-619	114	20	functional	functional	ADJ
ejpam-619	114	21	,	,	PUNCT
ejpam-619	114	22	we	we	PRON
ejpam-619	114	23	obtain	obtain	VERB
ejpam-619	114	24	φ(x)−φ(u	φ(x)−φ(u	NOUN
ejpam-619	114	25	)	)	PUNCT
ejpam-619	114	26	≧	≧	X
ejpam-619	115	1	f	f	PROPN
ejpam-619	115	2	�	�	PROPN
ejpam-619	115	3	x	x	PROPN
ejpam-619	115	4	,	,	PUNCT
ejpam-619	115	5	u;α(x	u;α(x	PROPN
ejpam-619	115	6	,	,	PUNCT
ejpam-619	115	7	u)[∇φ(u	u)[∇φ(u	NOUN
ejpam-619	115	8	)	)	PUNCT
ejpam-619	116	1	+	+	NUM
ejpam-619	116	2	∇vt	∇vt	NOUN
ejpam-619	116	3	h(u	h(u	PROPN
ejpam-619	116	4	)	)	PUNCT
ejpam-619	116	5	]	]	PUNCT
ejpam-619	116	6	�	�	PROPN
ejpam-619	117	1	+	+	NOUN
ejpam-619	117	2	ρ1d2(x	ρ1d2(x	PROPN
ejpam-619	117	3	,	,	PUNCT
ejpam-619	117	4	u	u	NOUN
ejpam-619	117	5	)	)	PUNCT
ejpam-619	117	6	+	+	NOUN
ejpam-619	117	7	ρ2d2(x	ρ2d2(x	X
ejpam-619	117	8	,	,	PUNCT
ejpam-619	117	9	u	u	NOUN
ejpam-619	117	10	)	)	PUNCT
ejpam-619	117	11	which	which	PRON
ejpam-619	117	12	in	in	ADP
ejpam-619	117	13	view	view	NOUN
ejpam-619	117	14	of	of	ADP
ejpam-619	117	15	(	(	PUNCT
ejpam-619	117	16	8)	8)	NUM
ejpam-619	117	17	implies	imply	VERB
ejpam-619	117	18	φ(x)−φ(u	φ(x)−φ(u	NOUN
ejpam-619	117	19	)	)	PUNCT
ejpam-619	117	20	≧	≧	X
ejpam-619	117	21	(	(	PUNCT
ejpam-619	117	22	ρ1+ρ2)d	ρ1+ρ2)d	PROPN
ejpam-619	117	23	2(x	2(x	NUM
ejpam-619	117	24	,	,	PUNCT
ejpam-619	117	25	u	u	NOUN
ejpam-619	117	26	)	)	PUNCT
ejpam-619	117	27	.	.	PUNCT
ejpam-619	118	1	using	use	VERB
ejpam-619	118	2	hypothesis	hypothesis	NOUN
ejpam-619	118	3	(	(	PUNCT
ejpam-619	118	4	iii	iii	NOUN
ejpam-619	118	5	)	)	PUNCT
ejpam-619	118	6	in	in	ADP
ejpam-619	118	7	the	the	DET
ejpam-619	118	8	above	above	ADJ
ejpam-619	118	9	inequality	inequality	NOUN
ejpam-619	118	10	,	,	PUNCT
ejpam-619	118	11	we	we	PRON
ejpam-619	118	12	get	get	VERB
ejpam-619	118	13	φ(x)≧	φ(x)≧	ADJ
ejpam-619	118	14	φ(u	φ(u	NOUN
ejpam-619	118	15	)	)	PUNCT
ejpam-619	118	16	.	.	PUNCT
ejpam-619	119	1	remark	remark	PROPN
ejpam-619	119	2	3	3	NUM
ejpam-619	119	3	.	.	PUNCT
ejpam-619	120	1	a	a	DET
ejpam-619	120	2	constraint	constraint	NOUN
ejpam-619	120	3	qualification	qualification	NOUN
ejpam-619	120	4	is	be	AUX
ejpam-619	120	5	not	not	PART
ejpam-619	120	6	required	require	VERB
ejpam-619	120	7	to	to	PART
ejpam-619	120	8	establish	establish	VERB
ejpam-619	120	9	weak	weak	ADJ
ejpam-619	120	10	duality	duality	NOUN
ejpam-619	120	11	.	.	PUNCT
ejpam-619	121	1	it	it	PRON
ejpam-619	121	2	has	have	AUX
ejpam-619	121	3	been	be	AUX
ejpam-619	121	4	erroneously	erroneously	ADV
ejpam-619	121	5	assumed	assume	VERB
ejpam-619	121	6	in	in	ADP
ejpam-619	121	7	theorem	theorem	NOUN
ejpam-619	121	8	3.4	3.4	NUM
ejpam-619	121	9	in	in	ADP
ejpam-619	121	10	[	[	X
ejpam-619	121	11	4	4	NUM
ejpam-619	121	12	]	]	PUNCT
ejpam-619	121	13	.	.	PUNCT
ejpam-619	122	1	theorem	theorem	ADJ
ejpam-619	122	2	3	3	NUM
ejpam-619	122	3	(	(	PUNCT
ejpam-619	122	4	strong	strong	ADJ
ejpam-619	122	5	duality	duality	NOUN
ejpam-619	122	6	)	)	PUNCT
ejpam-619	122	7	.	.	PUNCT
ejpam-619	123	1	let	let	VERB
ejpam-619	123	2	x̄	x̄	PRON
ejpam-619	123	3	be	be	AUX
ejpam-619	123	4	an	an	DET
ejpam-619	123	5	optimal	optimal	ADJ
ejpam-619	123	6	solution	solution	NOUN
ejpam-619	123	7	of	of	ADP
ejpam-619	123	8	the	the	DET
ejpam-619	123	9	problem	problem	NOUN
ejpam-619	123	10	(	(	PUNCT
ejpam-619	123	11	np	np	INTJ
ejpam-619	123	12	)	)	PUNCT
ejpam-619	123	13	and	and	CCONJ
ejpam-619	123	14	let	let	VERB
ejpam-619	123	15	h	h	PRON
ejpam-619	123	16	satisfy	satisfy	VERB
ejpam-619	123	17	a	a	DET
ejpam-619	123	18	constraint	constraint	NOUN
ejpam-619	123	19	qualification	qualification	NOUN
ejpam-619	123	20	.	.	PUNCT
ejpam-619	124	1	further	far	ADV
ejpam-619	124	2	,	,	PUNCT
ejpam-619	124	3	let	let	VERB
ejpam-619	124	4	theorem	theorem	VERB
ejpam-619	124	5	2	2	NUM
ejpam-619	124	6	hold	hold	NOUN
ejpam-619	124	7	for	for	ADP
ejpam-619	124	8	the	the	DET
ejpam-619	124	9	feasible	feasible	ADJ
ejpam-619	124	10	solution	solution	NOUN
ejpam-619	124	11	x̄	x̄	PROPN
ejpam-619	124	12	of	of	ADP
ejpam-619	124	13	(	(	PUNCT
ejpam-619	124	14	np	np	INTJ
ejpam-619	124	15	)	)	PUNCT
ejpam-619	124	16	and	and	CCONJ
ejpam-619	124	17	all	all	DET
ejpam-619	124	18	feasible	feasible	ADJ
ejpam-619	124	19	solutions	solution	NOUN
ejpam-619	124	20	(	(	PUNCT
ejpam-619	124	21	u	u	NOUN
ejpam-619	124	22	,	,	PUNCT
ejpam-619	124	23	v	v	NOUN
ejpam-619	124	24	)	)	PUNCT
ejpam-619	124	25	of	of	ADP
ejpam-619	124	26	(	(	PUNCT
ejpam-619	124	27	md	md	PROPN
ejpam-619	124	28	)	)	PUNCT
ejpam-619	124	29	.	.	PUNCT
ejpam-619	125	1	then	then	ADV
ejpam-619	125	2	there	there	PRON
ejpam-619	125	3	exists	exist	VERB
ejpam-619	125	4	a	a	DET
ejpam-619	125	5	v̄	v̄	NOUN
ejpam-619	125	6	∈	∈	PROPN
ejpam-619	125	7	rm	rm	NOUN
ejpam-619	125	8	+	+	CCONJ
ejpam-619	125	9	such	such	ADJ
ejpam-619	125	10	that	that	SCONJ
ejpam-619	125	11	(	(	PUNCT
ejpam-619	125	12	x̄	x̄	NOUN
ejpam-619	125	13	,	,	PUNCT
ejpam-619	125	14	v̄	v̄	PROPN
ejpam-619	125	15	)	)	PUNCT
ejpam-619	125	16	is	be	AUX
ejpam-619	125	17	an	an	DET
ejpam-619	125	18	optimal	optimal	ADJ
ejpam-619	125	19	solution	solution	NOUN
ejpam-619	125	20	of	of	ADP
ejpam-619	125	21	(	(	PUNCT
ejpam-619	125	22	md	md	PROPN
ejpam-619	125	23	)	)	PUNCT
ejpam-619	125	24	.	.	PUNCT
ejpam-619	126	1	proof	proof	NOUN
ejpam-619	126	2	.	.	PUNCT
ejpam-619	127	1	since	since	SCONJ
ejpam-619	127	2	x̄	x̄	PRON
ejpam-619	127	3	is	be	AUX
ejpam-619	127	4	an	an	DET
ejpam-619	127	5	optimal	optimal	ADJ
ejpam-619	127	6	solution	solution	NOUN
ejpam-619	127	7	for	for	ADP
ejpam-619	127	8	the	the	DET
ejpam-619	127	9	problem	problem	NOUN
ejpam-619	127	10	(	(	PUNCT
ejpam-619	127	11	np	np	INTJ
ejpam-619	127	12	)	)	PUNCT
ejpam-619	127	13	and	and	CCONJ
ejpam-619	127	14	h	h	NOUN
ejpam-619	127	15	satisfies	satisfy	VERB
ejpam-619	127	16	a	a	DET
ejpam-619	127	17	constraint	constraint	NOUN
ejpam-619	127	18	qualification	qualification	NOUN
ejpam-619	127	19	,	,	PUNCT
ejpam-619	127	20	by	by	ADP
ejpam-619	127	21	proposition	proposition	NOUN
ejpam-619	127	22	1	1	NUM
ejpam-619	127	23	there	there	ADV
ejpam-619	127	24	exists	exist	VERB
ejpam-619	127	25	a	a	DET
ejpam-619	127	26	v̄	v̄	NOUN
ejpam-619	127	27	∈	∈	PROPN
ejpam-619	127	28	rm	rm	NOUN
ejpam-619	127	29	+	+	CCONJ
ejpam-619	127	30	such	such	ADJ
ejpam-619	127	31	that	that	SCONJ
ejpam-619	127	32	the	the	DET
ejpam-619	127	33	kuhn	kuhn	PROPN
ejpam-619	127	34	-	-	PUNCT
ejpam-619	127	35	tucker	tucker	PROPN
ejpam-619	127	36	conditions	condition	NOUN
ejpam-619	127	37	,	,	PUNCT
ejpam-619	127	38	(	(	PUNCT
ejpam-619	127	39	1	1	X
ejpam-619	127	40	)	)	PUNCT
ejpam-619	127	41	(	(	PUNCT
ejpam-619	127	42	3	3	X
ejpam-619	127	43	)	)	PUNCT
ejpam-619	127	44	hold	hold	NOUN
ejpam-619	127	45	.	.	PUNCT
ejpam-619	128	1	hence	hence	ADV
ejpam-619	128	2	(	(	PUNCT
ejpam-619	128	3	x̄	x̄	NOUN
ejpam-619	128	4	,	,	PUNCT
ejpam-619	128	5	v̄	v̄	NOUN
ejpam-619	128	6	)	)	PUNCT
ejpam-619	128	7	is	be	AUX
ejpam-619	128	8	feasible	feasible	ADJ
ejpam-619	128	9	for	for	ADP
ejpam-619	128	10	(	(	PUNCT
ejpam-619	128	11	md	md	PROPN
ejpam-619	128	12	)	)	PUNCT
ejpam-619	128	13	.	.	PUNCT
ejpam-619	129	1	now	now	ADV
ejpam-619	129	2	let	let	VERB
ejpam-619	129	3	(	(	PUNCT
ejpam-619	129	4	u	u	NOUN
ejpam-619	129	5	,	,	PUNCT
ejpam-619	129	6	v	v	NOUN
ejpam-619	129	7	)	)	PUNCT
ejpam-619	129	8	be	be	VERB
ejpam-619	129	9	any	any	DET
ejpam-619	129	10	feasible	feasible	ADJ
ejpam-619	129	11	solution	solution	NOUN
ejpam-619	129	12	of	of	ADP
ejpam-619	129	13	(	(	PUNCT
ejpam-619	129	14	md	md	PROPN
ejpam-619	129	15	)	)	PUNCT
ejpam-619	129	16	.	.	PUNCT
ejpam-619	130	1	then	then	ADV
ejpam-619	130	2	by	by	ADP
ejpam-619	130	3	weak	weak	ADJ
ejpam-619	130	4	duality	duality	NOUN
ejpam-619	130	5	(	(	PUNCT
ejpam-619	130	6	theorem	theorem	NOUN
ejpam-619	130	7	2	2	NUM
ejpam-619	130	8	)	)	PUNCT
ejpam-619	130	9	,	,	PUNCT
ejpam-619	130	10	we	we	PRON
ejpam-619	130	11	have	have	VERB
ejpam-619	130	12	φ	φ	NUM
ejpam-619	130	13	(	(	PUNCT
ejpam-619	130	14	x̄)≧	x̄)≧	ADJ
ejpam-619	130	15	φ(u	φ(u	NOUN
ejpam-619	130	16	)	)	PUNCT
ejpam-619	130	17	.	.	PUNCT
ejpam-619	131	1	therefore	therefore	ADV
ejpam-619	131	2	(	(	PUNCT
ejpam-619	131	3	x̄	x̄	NOUN
ejpam-619	131	4	,	,	PUNCT
ejpam-619	131	5	v̄	v̄	NOUN
ejpam-619	131	6	)	)	PUNCT
ejpam-619	131	7	in	in	ADP
ejpam-619	131	8	an	an	DET
ejpam-619	131	9	optimal	optimal	ADJ
ejpam-619	131	10	solution	solution	NOUN
ejpam-619	131	11	of	of	ADP
ejpam-619	131	12	(	(	PUNCT
ejpam-619	131	13	md	md	PROPN
ejpam-619	131	14	)	)	PUNCT
ejpam-619	131	15	.	.	PUNCT
ejpam-619	132	1	t.	t.	PROPN
ejpam-619	132	2	gulati	gulati	PROPN
ejpam-619	132	3	,	,	PUNCT
ejpam-619	132	4	h.	h.	PROPN
ejpam-619	132	5	saini	saini	PROPN
ejpam-619	132	6	/	/	SYM
ejpam-619	132	7	eur	eur	PROPN
ejpam-619	132	8	.	.	PUNCT
ejpam-619	133	1	j.	j.	PROPN
ejpam-619	133	2	pure	pure	PROPN
ejpam-619	133	3	appl	appl	PROPN
ejpam-619	133	4	.	.	PROPN
ejpam-619	133	5	math	math	PROPN
ejpam-619	133	6	,	,	PUNCT
ejpam-619	133	7	4	4	NUM
ejpam-619	133	8	(	(	PUNCT
ejpam-619	133	9	2011	2011	NUM
ejpam-619	133	10	)	)	PUNCT
ejpam-619	133	11	,	,	PUNCT
ejpam-619	133	12	266	266	NUM
ejpam-619	133	13	-	-	SYM
ejpam-619	133	14	275	275	NUM
ejpam-619	133	15	271	271	NUM
ejpam-619	133	16	5	5	NUM
ejpam-619	133	17	.	.	PUNCT
ejpam-619	133	18	application	application	NOUN
ejpam-619	133	19	in	in	ADP
ejpam-619	133	20	fractional	fractional	ADJ
ejpam-619	133	21	programming	programming	NOUN
ejpam-619	133	22	if	if	SCONJ
ejpam-619	133	23	φ	φ	PROPN
ejpam-619	133	24	:	:	PUNCT
ejpam-619	133	25	x	x	X
ejpam-619	133	26	→	→	SYM
ejpam-619	133	27	r	r	NOUN
ejpam-619	133	28	is	be	AUX
ejpam-619	133	29	defined	define	VERB
ejpam-619	133	30	by	by	ADP
ejpam-619	133	31	φ(x	φ(x	NOUN
ejpam-619	133	32	)	)	PUNCT
ejpam-619	133	33	=	=	SYM
ejpam-619	134	1	f	f	X
ejpam-619	134	2	(	(	PUNCT
ejpam-619	134	3	x	x	NOUN
ejpam-619	134	4	)	)	PUNCT
ejpam-619	134	5	g(x	g(x	NOUN
ejpam-619	134	6	)	)	PUNCT
ejpam-619	134	7	,	,	PUNCT
ejpam-619	134	8	where	where	SCONJ
ejpam-619	134	9	f	f	PROPN
ejpam-619	134	10	,	,	PUNCT
ejpam-619	134	11	g	g	NOUN
ejpam-619	134	12	:	:	PUNCT
ejpam-619	134	13	x	x	SYM
ejpam-619	134	14	→	→	SYM
ejpam-619	134	15	r	r	NOUN
ejpam-619	134	16	,	,	PUNCT
ejpam-619	134	17	f	f	PROPN
ejpam-619	134	18	(	(	PUNCT
ejpam-619	134	19	x	x	X
ejpam-619	134	20	)	)	PUNCT
ejpam-619	134	21	≧	≧	NOUN
ejpam-619	134	22	0	0	NUM
ejpam-619	134	23	and	and	CCONJ
ejpam-619	134	24	g(x	g(x	NOUN
ejpam-619	134	25	)	)	PUNCT
ejpam-619	134	26	>	>	X
ejpam-619	134	27	0	0	PUNCT
ejpam-619	135	1	on	on	ADP
ejpam-619	135	2	x	x	X
ejpam-619	135	3	,	,	PUNCT
ejpam-619	135	4	then	then	ADV
ejpam-619	135	5	the	the	DET
ejpam-619	135	6	nonlinear	nonlinear	ADJ
ejpam-619	135	7	programming	programming	NOUN
ejpam-619	135	8	problem	problem	NOUN
ejpam-619	135	9	(	(	PUNCT
ejpam-619	135	10	np	np	INTJ
ejpam-619	135	11	)	)	PUNCT
ejpam-619	135	12	becomes	become	VERB
ejpam-619	135	13	the	the	DET
ejpam-619	135	14	following	follow	VERB
ejpam-619	135	15	fractional	fractional	ADJ
ejpam-619	135	16	programming	programming	NOUN
ejpam-619	135	17	problem	problem	NOUN
ejpam-619	135	18	(	(	PUNCT
ejpam-619	135	19	fp	fp	X
ejpam-619	135	20	):	):	PUNCT
ejpam-619	135	21	(	(	PUNCT
ejpam-619	135	22	fp	fp	X
ejpam-619	135	23	)	)	PUNCT
ejpam-619	135	24	minimize	minimize	VERB
ejpam-619	135	25	f	f	X
ejpam-619	135	26	(	(	PUNCT
ejpam-619	135	27	x	x	NOUN
ejpam-619	135	28	)	)	PUNCT
ejpam-619	135	29	g(x	g(x	NOUN
ejpam-619	135	30	)	)	PUNCT
ejpam-619	135	31	subject	subject	NOUN
ejpam-619	135	32	to	to	ADP
ejpam-619	135	33	h(x)≦	h(x)≦	NOUN
ejpam-619	135	34	0	0	NUM
ejpam-619	135	35	,	,	PUNCT
ejpam-619	135	36	x	x	X
ejpam-619	135	37	∈	∈	NOUN
ejpam-619	135	38	x	x	X
ejpam-619	135	39	.	.	PUNCT
ejpam-619	136	1	we	we	PRON
ejpam-619	136	2	now	now	ADV
ejpam-619	136	3	prove	prove	VERB
ejpam-619	136	4	the	the	DET
ejpam-619	136	5	following	follow	VERB
ejpam-619	136	6	result	result	NOUN
ejpam-619	136	7	,	,	PUNCT
ejpam-619	136	8	which	which	PRON
ejpam-619	136	9	gives	give	VERB
ejpam-619	136	10	higher	high	ADJ
ejpam-619	136	11	-	-	PUNCT
ejpam-619	136	12	order	order	NOUN
ejpam-619	136	13	(	(	PUNCT
ejpam-619	136	14	f	f	NOUN
ejpam-619	136	15	,	,	PUNCT
ejpam-619	136	16	ᾱ	ᾱ	NOUN
ejpam-619	136	17	,	,	PUNCT
ejpam-619	136	18	β̄	β̄	NOUN
ejpam-619	136	19	,	,	PUNCT
ejpam-619	136	20	ρ	ρ	NOUN
ejpam-619	136	21	,	,	PUNCT
ejpam-619	136	22	d̄)-convexity	d̄)-convexity	PROPN
ejpam-619	136	23	of	of	ADP
ejpam-619	136	24	the	the	DET
ejpam-619	136	25	ratio	ratio	NOUN
ejpam-619	136	26	function	function	NOUN
ejpam-619	136	27	f	f	PROPN
ejpam-619	136	28	(	(	PUNCT
ejpam-619	136	29	x)/g(x	x)/g(x	PROPN
ejpam-619	136	30	)	)	PUNCT
ejpam-619	136	31	.	.	PUNCT
ejpam-619	137	1	theorem	theorem	ADJ
ejpam-619	137	2	4	4	NUM
ejpam-619	137	3	.	.	PUNCT
ejpam-619	138	1	let	let	VERB
ejpam-619	138	2	f	f	PROPN
ejpam-619	138	3	(	(	PUNCT
ejpam-619	138	4	x	x	NOUN
ejpam-619	138	5	)	)	PUNCT
ejpam-619	138	6	and	and	CCONJ
ejpam-619	138	7	−g(x	−g(x	X
ejpam-619	138	8	)	)	PUNCT
ejpam-619	138	9	be	be	AUX
ejpam-619	138	10	higher	high	ADJ
ejpam-619	138	11	-	-	PUNCT
ejpam-619	138	12	order	order	NOUN
ejpam-619	138	13	(	(	PUNCT
ejpam-619	138	14	f	f	X
ejpam-619	138	15	,	,	PUNCT
ejpam-619	138	16	α	α	PROPN
ejpam-619	138	17	,	,	PUNCT
ejpam-619	138	18	β	β	PROPN
ejpam-619	138	19	,	,	PUNCT
ejpam-619	138	20	ρ	ρ	PROPN
ejpam-619	138	21	,	,	PUNCT
ejpam-619	138	22	d)-convex	d)-convex	PUNCT
ejpam-619	138	23	at	at	ADP
ejpam-619	138	24	x̄	x̄	NOUN
ejpam-619	138	25	with	with	ADP
ejpam-619	138	26	respect	respect	NOUN
ejpam-619	138	27	to	to	ADP
ejpam-619	138	28	the	the	DET
ejpam-619	138	29	same	same	ADJ
ejpam-619	138	30	function	function	NOUN
ejpam-619	138	31	k.	k.	PROPN
ejpam-619	139	1	then	then	ADV
ejpam-619	139	2	the	the	DET
ejpam-619	139	3	fractional	fractional	ADJ
ejpam-619	139	4	function	function	NOUN
ejpam-619	139	5	f	f	PROPN
ejpam-619	139	6	(	(	PUNCT
ejpam-619	139	7	x	x	NOUN
ejpam-619	139	8	)	)	PUNCT
ejpam-619	139	9	g(x	g(x	NOUN
ejpam-619	139	10	)	)	PUNCT
ejpam-619	139	11	is	be	AUX
ejpam-619	139	12	higher	high	ADJ
ejpam-619	139	13	-	-	PUNCT
ejpam-619	139	14	order	order	NOUN
ejpam-619	139	15	(	(	PUNCT
ejpam-619	139	16	f	f	NOUN
ejpam-619	139	17	,	,	PUNCT
ejpam-619	139	18	ᾱ	ᾱ	NOUN
ejpam-619	139	19	,	,	PUNCT
ejpam-619	139	20	β̄	β̄	NOUN
ejpam-619	139	21	,	,	PUNCT
ejpam-619	139	22	ρ	ρ	PROPN
ejpam-619	139	23	,	,	PUNCT
ejpam-619	139	24	d̄)-convex	d̄)-convex	PROPN
ejpam-619	139	25	at	at	ADP
ejpam-619	139	26	x̄	x̄	NOUN
ejpam-619	139	27	with	with	ADP
ejpam-619	139	28	respect	respect	NOUN
ejpam-619	139	29	to	to	ADP
ejpam-619	139	30	k̄	k̄	PROPN
ejpam-619	139	31	,	,	PUNCT
ejpam-619	139	32	where	where	SCONJ
ejpam-619	139	33	ᾱ(x	ᾱ(x	PROPN
ejpam-619	139	34	,	,	PUNCT
ejpam-619	139	35	x̄	x̄	PROPN
ejpam-619	139	36	)	)	PUNCT
ejpam-619	139	37	=	=	SYM
ejpam-619	139	38	α(x	α(x	PROPN
ejpam-619	139	39	,	,	PUNCT
ejpam-619	139	40	x̄	x̄	PROPN
ejpam-619	139	41	)	)	PUNCT
ejpam-619	139	42	g	g	PROPN
ejpam-619	139	43	(	(	PUNCT
ejpam-619	139	44	x̄	x̄	NOUN
ejpam-619	139	45	)	)	PUNCT
ejpam-619	139	46	g(x	g(x	NOUN
ejpam-619	139	47	)	)	PUNCT
ejpam-619	139	48	β̄(x	β̄(x	NOUN
ejpam-619	139	49	,	,	PUNCT
ejpam-619	139	50	x̄	x̄	X
ejpam-619	139	51	)	)	PUNCT
ejpam-619	140	1	=	=	NOUN
ejpam-619	140	2	β(x	β(x	NOUN
ejpam-619	140	3	,	,	PUNCT
ejpam-619	140	4	x̄	x̄	NUM
ejpam-619	140	5	)	)	PUNCT
ejpam-619	140	6	g	g	PROPN
ejpam-619	140	7	(	(	PUNCT
ejpam-619	140	8	x̄	x̄	NOUN
ejpam-619	140	9	)	)	PUNCT
ejpam-619	140	10	g(x	g(x	NOUN
ejpam-619	140	11	)	)	PUNCT
ejpam-619	140	12	,	,	PUNCT
ejpam-619	140	13	k̄	k̄	PROPN
ejpam-619	140	14	(	(	PUNCT
ejpam-619	140	15	x̄	x̄	NOUN
ejpam-619	140	16	,	,	PUNCT
ejpam-619	140	17	p	p	X
ejpam-619	140	18	)	)	PUNCT
ejpam-619	140	19	=	=	SYM
ejpam-619	140	20	�	�	PROPN
ejpam-619	140	21	1	1	NUM
ejpam-619	140	22	g	g	PROPN
ejpam-619	140	23	(	(	PUNCT
ejpam-619	140	24	x̄	x̄	PROPN
ejpam-619	140	25	)	)	PUNCT
ejpam-619	141	1	+	+	CCONJ
ejpam-619	141	2	f	f	X
ejpam-619	141	3	(	(	PUNCT
ejpam-619	141	4	x̄	x̄	PROPN
ejpam-619	141	5	)	)	PUNCT
ejpam-619	141	6	g2	g2	PROPN
ejpam-619	141	7	(	(	PUNCT
ejpam-619	141	8	x̄	x̄	PROPN
ejpam-619	141	9	)	)	PUNCT
ejpam-619	141	10	�	�	PROPN
ejpam-619	142	1	k	k	PROPN
ejpam-619	142	2	(	(	PUNCT
ejpam-619	142	3	x̄	x̄	NOUN
ejpam-619	142	4	,	,	PUNCT
ejpam-619	142	5	p	p	NOUN
ejpam-619	142	6	)	)	PUNCT
ejpam-619	142	7	,	,	PUNCT
ejpam-619	142	8	d̄(x	d̄(x	PROPN
ejpam-619	142	9	,	,	PUNCT
ejpam-619	142	10	x̄	x̄	X
ejpam-619	142	11	)	)	PUNCT
ejpam-619	142	12	=	=	SYM
ejpam-619	142	13	�	�	PROPN
ejpam-619	142	14	1	1	NUM
ejpam-619	142	15	g(x	g(x	NOUN
ejpam-619	142	16	)	)	PUNCT
ejpam-619	143	1	+	+	CCONJ
ejpam-619	143	2	f	f	X
ejpam-619	143	3	(	(	PUNCT
ejpam-619	143	4	x̄	x̄	PROPN
ejpam-619	143	5	)	)	PUNCT
ejpam-619	143	6	g(x)g	g(x)g	PROPN
ejpam-619	143	7	(	(	PUNCT
ejpam-619	143	8	x̄	x̄	NOUN
ejpam-619	143	9	)	)	PUNCT
ejpam-619	143	10	�	�	PROPN
ejpam-619	143	11	1	1	NUM
ejpam-619	143	12	2	2	NUM
ejpam-619	143	13	d(x	d(x	NOUN
ejpam-619	143	14	,	,	PUNCT
ejpam-619	143	15	x̄	x̄	PROPN
ejpam-619	143	16	)	)	PUNCT
ejpam-619	143	17	.	.	PUNCT
ejpam-619	144	1	proof	proof	NOUN
ejpam-619	144	2	.	.	PUNCT
ejpam-619	145	1	since	since	SCONJ
ejpam-619	145	2	f	f	PROPN
ejpam-619	145	3	(	(	PUNCT
ejpam-619	145	4	x	x	NOUN
ejpam-619	145	5	)	)	PUNCT
ejpam-619	145	6	and	and	CCONJ
ejpam-619	145	7	−g(x	−g(x	X
ejpam-619	145	8	)	)	PUNCT
ejpam-619	145	9	are	be	AUX
ejpam-619	145	10	higher	high	ADJ
ejpam-619	145	11	-	-	PUNCT
ejpam-619	145	12	order	order	NOUN
ejpam-619	145	13	(	(	PUNCT
ejpam-619	145	14	f	f	X
ejpam-619	145	15	,	,	PUNCT
ejpam-619	145	16	α	α	PROPN
ejpam-619	145	17	,	,	PUNCT
ejpam-619	145	18	β	β	PROPN
ejpam-619	145	19	,	,	PUNCT
ejpam-619	145	20	ρ	ρ	PROPN
ejpam-619	145	21	,	,	PUNCT
ejpam-619	145	22	d)-convex	d)-convex	PUNCT
ejpam-619	145	23	at	at	ADP
ejpam-619	145	24	x̄	x̄	NOUN
ejpam-619	145	25	with	with	ADP
ejpam-619	145	26	respect	respect	NOUN
ejpam-619	145	27	to	to	ADP
ejpam-619	145	28	the	the	DET
ejpam-619	145	29	same	same	ADJ
ejpam-619	145	30	function	function	NOUN
ejpam-619	145	31	k	k	NOUN
ejpam-619	145	32	,	,	PUNCT
ejpam-619	145	33	we	we	PRON
ejpam-619	145	34	have	have	VERB
ejpam-619	145	35	f	f	X
ejpam-619	145	36	(	(	PUNCT
ejpam-619	145	37	x)−	x)−	PROPN
ejpam-619	145	38	f	f	PROPN
ejpam-619	145	39	(	(	PUNCT
ejpam-619	145	40	x̄	x̄	PROPN
ejpam-619	145	41	)	)	PUNCT
ejpam-619	145	42	≧	≧	PUNCT
ejpam-619	146	1	f(x	f(x	PROPN
ejpam-619	146	2	,	,	PUNCT
ejpam-619	146	3	x̄;α(x	x̄;α(x	PROPN
ejpam-619	146	4	,	,	PUNCT
ejpam-619	146	5	x̄){∇	x̄){∇	PROPN
ejpam-619	146	6	f	f	PROPN
ejpam-619	147	1	(	(	PUNCT
ejpam-619	147	2	x̄	x̄	PROPN
ejpam-619	147	3	)	)	PUNCT
ejpam-619	148	1	+	+	X
ejpam-619	148	2	∇pk	∇pk	NOUN
ejpam-619	148	3	(	(	PUNCT
ejpam-619	148	4	x̄	x̄	NOUN
ejpam-619	148	5	,	,	PUNCT
ejpam-619	148	6	p	p	NOUN
ejpam-619	148	7	)	)	PUNCT
ejpam-619	148	8	}	}	PUNCT
ejpam-619	148	9	)	)	PUNCT
ejpam-619	148	10	+	+	NUM
ejpam-619	148	11	β(x	β(x	NOUN
ejpam-619	148	12	,	,	PUNCT
ejpam-619	148	13	x̄){k	x̄){k	NOUN
ejpam-619	148	14	(	(	PUNCT
ejpam-619	148	15	x̄	x̄	NOUN
ejpam-619	148	16	,	,	PUNCT
ejpam-619	148	17	p)−	p)−	PROPN
ejpam-619	148	18	pt∇pk	pt∇pk	PROPN
ejpam-619	148	19	(	(	PUNCT
ejpam-619	148	20	x̄	x̄	NOUN
ejpam-619	148	21	,	,	PUNCT
ejpam-619	148	22	p)}+ρd2(x	p)}+ρd2(x	PROPN
ejpam-619	148	23	,	,	PUNCT
ejpam-619	148	24	x̄	x̄	PROPN
ejpam-619	148	25	)	)	PUNCT
ejpam-619	148	26	and	and	CCONJ
ejpam-619	148	27	−g(x)+	−g(x)+	PROPN
ejpam-619	148	28	g	g	PROPN
ejpam-619	148	29	(	(	PUNCT
ejpam-619	148	30	x̄)≧	x̄)≧	PROPN
ejpam-619	148	31	f(x	f(x	PROPN
ejpam-619	148	32	,	,	PUNCT
ejpam-619	148	33	x̄	x̄	PROPN
ejpam-619	148	34	;	;	PUNCT
ejpam-619	148	35	α(x	α(x	PROPN
ejpam-619	148	36	,	,	PUNCT
ejpam-619	148	37	x̄){−∇g	x̄){−∇g	PROPN
ejpam-619	148	38	(	(	PUNCT
ejpam-619	148	39	x̄	x̄	PROPN
ejpam-619	148	40	)	)	PUNCT
ejpam-619	149	1	+	+	X
ejpam-619	149	2	∇pk	∇pk	NOUN
ejpam-619	149	3	(	(	PUNCT
ejpam-619	149	4	x̄	x̄	NOUN
ejpam-619	149	5	,	,	PUNCT
ejpam-619	149	6	p	p	NOUN
ejpam-619	149	7	)	)	PUNCT
ejpam-619	149	8	}	}	PUNCT
ejpam-619	149	9	)	)	PUNCT
ejpam-619	149	10	+	+	NUM
ejpam-619	149	11	β(x	β(x	NOUN
ejpam-619	149	12	,	,	PUNCT
ejpam-619	149	13	x̄){k	x̄){k	NOUN
ejpam-619	149	14	(	(	PUNCT
ejpam-619	149	15	x̄	x̄	NOUN
ejpam-619	149	16	,	,	PUNCT
ejpam-619	149	17	p)−	p)−	PROPN
ejpam-619	149	18	pt∇pk	pt∇pk	PROPN
ejpam-619	149	19	(	(	PUNCT
ejpam-619	149	20	x̄	x̄	NOUN
ejpam-619	149	21	,	,	PUNCT
ejpam-619	149	22	p)}+ρd2(x	p)}+ρd2(x	PROPN
ejpam-619	149	23	,	,	PUNCT
ejpam-619	149	24	x̄	x̄	PROPN
ejpam-619	149	25	)	)	PUNCT
ejpam-619	149	26	.	.	PUNCT
ejpam-619	150	1	also	also	ADV
ejpam-619	150	2	f	f	X
ejpam-619	150	3	(	(	PUNCT
ejpam-619	150	4	x	x	NOUN
ejpam-619	150	5	)	)	PUNCT
ejpam-619	150	6	g(x	g(x	NOUN
ejpam-619	150	7	)	)	PUNCT
ejpam-619	151	1	−	−	PROPN
ejpam-619	151	2	f	f	PROPN
ejpam-619	151	3	(	(	PUNCT
ejpam-619	151	4	x̄	x̄	PROPN
ejpam-619	151	5	)	)	PUNCT
ejpam-619	151	6	g	g	PROPN
ejpam-619	151	7	(	(	PUNCT
ejpam-619	151	8	x̄	x̄	PROPN
ejpam-619	151	9	)	)	PUNCT
ejpam-619	151	10	=	=	SYM
ejpam-619	151	11	1	1	NUM
ejpam-619	151	12	g(x	g(x	NOUN
ejpam-619	151	13	)	)	PUNCT
ejpam-619	151	14	�	�	PROPN
ejpam-619	151	15	f	f	PROPN
ejpam-619	151	16	(	(	PUNCT
ejpam-619	151	17	x)−	x)−	PROPN
ejpam-619	151	18	f	f	PROPN
ejpam-619	151	19	(	(	PUNCT
ejpam-619	151	20	x̄	x̄	PROPN
ejpam-619	151	21	)	)	PUNCT
ejpam-619	151	22	�	�	PROPN
ejpam-619	152	1	+	+	NUM
ejpam-619	152	2	f	f	PROPN
ejpam-619	152	3	(	(	PUNCT
ejpam-619	152	4	x̄	x̄	PROPN
ejpam-619	152	5	)	)	PUNCT
ejpam-619	152	6	g(x)g	g(x)g	PROPN
ejpam-619	152	7	(	(	PUNCT
ejpam-619	152	8	x̄	x̄	NOUN
ejpam-619	152	9	)	)	PUNCT
ejpam-619	152	10	�	�	PROPN
ejpam-619	152	11	−g(x)+	−g(x)+	PROPN
ejpam-619	152	12	g	g	PROPN
ejpam-619	152	13	(	(	PUNCT
ejpam-619	152	14	x̄	x̄	PROPN
ejpam-619	152	15	)	)	PUNCT
ejpam-619	152	16	�	�	PROPN
ejpam-619	152	17	.	.	PUNCT
ejpam-619	153	1	t.	t.	PROPN
ejpam-619	153	2	gulati	gulati	PROPN
ejpam-619	153	3	,	,	PUNCT
ejpam-619	153	4	h.	h.	PROPN
ejpam-619	153	5	saini	saini	PROPN
ejpam-619	153	6	/	/	SYM
ejpam-619	153	7	eur	eur	PROPN
ejpam-619	153	8	.	.	PUNCT
ejpam-619	154	1	j.	j.	PROPN
ejpam-619	154	2	pure	pure	PROPN
ejpam-619	154	3	appl	appl	PROPN
ejpam-619	154	4	.	.	PROPN
ejpam-619	154	5	math	math	PROPN
ejpam-619	154	6	,	,	PUNCT
ejpam-619	154	7	4	4	NUM
ejpam-619	154	8	(	(	PUNCT
ejpam-619	154	9	2011	2011	NUM
ejpam-619	154	10	)	)	PUNCT
ejpam-619	154	11	,	,	PUNCT
ejpam-619	154	12	266	266	NUM
ejpam-619	154	13	-	-	SYM
ejpam-619	154	14	275	275	NUM
ejpam-619	154	15	272	272	NUM
ejpam-619	154	16	using	use	VERB
ejpam-619	154	17	the	the	DET
ejpam-619	154	18	above	above	ADJ
ejpam-619	154	19	inequalities	inequality	NOUN
ejpam-619	154	20	and	and	CCONJ
ejpam-619	154	21	sublinearity	sublinearity	NOUN
ejpam-619	154	22	of	of	ADP
ejpam-619	154	23	f	f	PROPN
ejpam-619	154	24	,	,	PUNCT
ejpam-619	154	25	we	we	PRON
ejpam-619	154	26	get	get	VERB
ejpam-619	154	27	f	f	PROPN
ejpam-619	154	28	(	(	PUNCT
ejpam-619	154	29	x	x	NOUN
ejpam-619	154	30	)	)	PUNCT
ejpam-619	154	31	g(x	g(x	NOUN
ejpam-619	154	32	)	)	PUNCT
ejpam-619	155	1	−	−	PROPN
ejpam-619	155	2	f	f	PROPN
ejpam-619	155	3	(	(	PUNCT
ejpam-619	155	4	x̄	x̄	PROPN
ejpam-619	155	5	)	)	PUNCT
ejpam-619	155	6	g	g	PROPN
ejpam-619	155	7	(	(	PUNCT
ejpam-619	155	8	x̄	x̄	PROPN
ejpam-619	155	9	)	)	PUNCT
ejpam-619	155	10	≧	≧	SYM
ejpam-619	156	1	1	1	NUM
ejpam-619	156	2	g(x	g(x	NOUN
ejpam-619	156	3	)	)	PUNCT
ejpam-619	156	4	f(x	f(x	PROPN
ejpam-619	156	5	,	,	PUNCT
ejpam-619	156	6	x̄;α(x	x̄;α(x	PROPN
ejpam-619	156	7	,	,	PUNCT
ejpam-619	156	8	x̄){∇	x̄){∇	PROPN
ejpam-619	156	9	f	f	PROPN
ejpam-619	156	10	(	(	PUNCT
ejpam-619	156	11	x̄)+∇pk	x̄)+∇pk	PROPN
ejpam-619	156	12	(	(	PUNCT
ejpam-619	156	13	x̄	x̄	NOUN
ejpam-619	156	14	,	,	PUNCT
ejpam-619	156	15	p	p	NOUN
ejpam-619	156	16	)	)	PUNCT
ejpam-619	156	17	}	}	PUNCT
ejpam-619	156	18	)	)	PUNCT
ejpam-619	157	1	+	+	CCONJ
ejpam-619	157	2	1	1	NUM
ejpam-619	157	3	g(x	g(x	NUM
ejpam-619	157	4	)	)	PUNCT
ejpam-619	157	5	�	�	PROPN
ejpam-619	157	6	β(x	β(x	NOUN
ejpam-619	157	7	,	,	PUNCT
ejpam-619	157	8	x̄){k	x̄){k	NOUN
ejpam-619	157	9	(	(	PUNCT
ejpam-619	157	10	x̄	x̄	NOUN
ejpam-619	157	11	,	,	PUNCT
ejpam-619	157	12	p)−	p)−	PROPN
ejpam-619	157	13	pt∇pk	pt∇pk	PROPN
ejpam-619	157	14	(	(	PUNCT
ejpam-619	157	15	x̄	x̄	NOUN
ejpam-619	157	16	,	,	PUNCT
ejpam-619	157	17	p)}+ρd2(x	p)}+ρd2(x	PROPN
ejpam-619	157	18	,	,	PUNCT
ejpam-619	157	19	x̄	x̄	ADJ
ejpam-619	157	20	)	)	PUNCT
ejpam-619	157	21	�	�	PROPN
ejpam-619	157	22	+	+	NUM
ejpam-619	157	23	f	f	PROPN
ejpam-619	157	24	(	(	PUNCT
ejpam-619	157	25	x̄	x̄	PROPN
ejpam-619	157	26	)	)	PUNCT
ejpam-619	157	27	g(x)g	g(x)g	PROPN
ejpam-619	157	28	(	(	PUNCT
ejpam-619	157	29	x̄	x̄	NOUN
ejpam-619	157	30	)	)	PUNCT
ejpam-619	157	31	f(x	f(x	PROPN
ejpam-619	157	32	,	,	PUNCT
ejpam-619	157	33	x̄	x̄	NOUN
ejpam-619	157	34	;	;	PUNCT
ejpam-619	157	35	α(x	α(x	PROPN
ejpam-619	157	36	,	,	PUNCT
ejpam-619	157	37	x̄){−∇g	x̄){−∇g	PROPN
ejpam-619	157	38	(	(	PUNCT
ejpam-619	157	39	x̄	x̄	PROPN
ejpam-619	157	40	)	)	PUNCT
ejpam-619	158	1	+	+	X
ejpam-619	158	2	∇pk	∇pk	NOUN
ejpam-619	158	3	(	(	PUNCT
ejpam-619	158	4	x̄	x̄	NOUN
ejpam-619	158	5	,	,	PUNCT
ejpam-619	158	6	p	p	NOUN
ejpam-619	158	7	)	)	PUNCT
ejpam-619	158	8	}	}	PUNCT
ejpam-619	158	9	)	)	PUNCT
ejpam-619	159	1	+	+	CCONJ
ejpam-619	159	2	f	f	X
ejpam-619	159	3	(	(	PUNCT
ejpam-619	159	4	x̄	x̄	PROPN
ejpam-619	159	5	)	)	PUNCT
ejpam-619	159	6	g(x)g	g(x)g	PROPN
ejpam-619	159	7	(	(	PUNCT
ejpam-619	159	8	x̄	x̄	ADJ
ejpam-619	159	9	)	)	PUNCT
ejpam-619	159	10	�	�	PROPN
ejpam-619	159	11	β(x	β(x	PROPN
ejpam-619	159	12	,	,	PUNCT
ejpam-619	159	13	x̄){k	x̄){k	NOUN
ejpam-619	159	14	(	(	PUNCT
ejpam-619	159	15	x̄	x̄	NOUN
ejpam-619	159	16	,	,	PUNCT
ejpam-619	159	17	p)−	p)−	PROPN
ejpam-619	159	18	pt∇pk	pt∇pk	PROPN
ejpam-619	159	19	(	(	PUNCT
ejpam-619	159	20	x̄	x̄	NOUN
ejpam-619	159	21	,	,	PUNCT
ejpam-619	159	22	p)}+ρd2(x	p)}+ρd2(x	PROPN
ejpam-619	159	23	,	,	PUNCT
ejpam-619	159	24	x̄	x̄	ADJ
ejpam-619	159	25	)	)	PUNCT
ejpam-619	159	26	�	�	PROPN
ejpam-619	159	27	.	.	PUNCT
ejpam-619	160	1	=	=	PUNCT
ejpam-619	160	2	f(x	f(x	PROPN
ejpam-619	160	3	,	,	PUNCT
ejpam-619	160	4	x̄	x̄	NOUN
ejpam-619	160	5	;	;	PUNCT
ejpam-619	160	6	α(x	α(x	PROPN
ejpam-619	160	7	,	,	PUNCT
ejpam-619	160	8	x̄	x̄	ADJ
ejpam-619	160	9	)	)	PUNCT
ejpam-619	160	10	g(x	g(x	NOUN
ejpam-619	160	11	)	)	PUNCT
ejpam-619	160	12	{	{	PUNCT
ejpam-619	160	13	∇	∇	X
ejpam-619	160	14	f	f	X
ejpam-619	160	15	(	(	PUNCT
ejpam-619	160	16	x̄	x̄	PROPN
ejpam-619	160	17	)	)	PUNCT
ejpam-619	161	1	+	+	X
ejpam-619	161	2	∇pk	∇pk	NOUN
ejpam-619	161	3	(	(	PUNCT
ejpam-619	161	4	x̄	x̄	NOUN
ejpam-619	161	5	,	,	PUNCT
ejpam-619	161	6	p	p	NOUN
ejpam-619	161	7	)	)	PUNCT
ejpam-619	161	8	}	}	PUNCT
ejpam-619	161	9	)	)	PUNCT
ejpam-619	162	1	+	+	ADP
ejpam-619	162	2	f(x	f(x	PROPN
ejpam-619	162	3	,	,	PUNCT
ejpam-619	162	4	x̄;α(x	x̄;α(x	X
ejpam-619	162	5	,	,	PUNCT
ejpam-619	162	6	x̄	x̄	PROPN
ejpam-619	162	7	)	)	PUNCT
ejpam-619	162	8	f	f	PROPN
ejpam-619	162	9	(	(	PUNCT
ejpam-619	162	10	x̄	x̄	PROPN
ejpam-619	162	11	)	)	PUNCT
ejpam-619	162	12	g(x)g	g(x)g	PROPN
ejpam-619	162	13	(	(	PUNCT
ejpam-619	162	14	x̄	x̄	PROPN
ejpam-619	162	15	)	)	PUNCT
ejpam-619	162	16	{	{	PUNCT
ejpam-619	162	17	−∇g	−∇g	PROPN
ejpam-619	162	18	(	(	PUNCT
ejpam-619	162	19	x̄	x̄	PROPN
ejpam-619	162	20	)	)	PUNCT
ejpam-619	163	1	+	+	X
ejpam-619	163	2	∇pk	∇pk	NOUN
ejpam-619	163	3	(	(	PUNCT
ejpam-619	163	4	x̄	x̄	NOUN
ejpam-619	163	5	,	,	PUNCT
ejpam-619	163	6	p	p	NOUN
ejpam-619	163	7	)	)	PUNCT
ejpam-619	163	8	}	}	PUNCT
ejpam-619	163	9	)	)	PUNCT
ejpam-619	163	10	+	+	NOUN
ejpam-619	163	11	β(x	β(x	NOUN
ejpam-619	163	12	,	,	PUNCT
ejpam-619	163	13	x̄	x̄	NUM
ejpam-619	163	14	)	)	PUNCT
ejpam-619	163	15	�	�	PROPN
ejpam-619	163	16	1	1	NUM
ejpam-619	163	17	g(x	g(x	NOUN
ejpam-619	163	18	)	)	PUNCT
ejpam-619	164	1	+	+	CCONJ
ejpam-619	164	2	f	f	X
ejpam-619	164	3	(	(	PUNCT
ejpam-619	164	4	x̄	x̄	PROPN
ejpam-619	164	5	)	)	PUNCT
ejpam-619	164	6	g(x)g	g(x)g	PROPN
ejpam-619	164	7	(	(	PUNCT
ejpam-619	164	8	x̄	x̄	NOUN
ejpam-619	164	9	)	)	PUNCT
ejpam-619	164	10	�	�	PROPN
ejpam-619	164	11	{	{	PUNCT
ejpam-619	164	12	k	k	X
ejpam-619	164	13	(	(	PUNCT
ejpam-619	164	14	x̄	x̄	NOUN
ejpam-619	164	15	,	,	PUNCT
ejpam-619	164	16	p)−	p)−	PROPN
ejpam-619	164	17	pt∇pk	pt∇pk	PROPN
ejpam-619	164	18	(	(	PUNCT
ejpam-619	164	19	x̄	x̄	NOUN
ejpam-619	164	20	,	,	PUNCT
ejpam-619	164	21	p	p	NOUN
ejpam-619	164	22	)	)	PUNCT
ejpam-619	164	23	}	}	PUNCT
ejpam-619	165	1	+	+	NOUN
ejpam-619	165	2	ρ	ρ	PROPN
ejpam-619	165	3	�	�	PROPN
ejpam-619	165	4	1	1	NUM
ejpam-619	165	5	g(x	g(x	NOUN
ejpam-619	165	6	)	)	PUNCT
ejpam-619	166	1	+	+	CCONJ
ejpam-619	166	2	f	f	X
ejpam-619	166	3	(	(	PUNCT
ejpam-619	166	4	x̄	x̄	PROPN
ejpam-619	166	5	)	)	PUNCT
ejpam-619	166	6	g(x)g	g(x)g	PROPN
ejpam-619	166	7	(	(	PUNCT
ejpam-619	166	8	x̄	x̄	NOUN
ejpam-619	166	9	)	)	PUNCT
ejpam-619	166	10	�	�	PROPN
ejpam-619	166	11	d2(x	d2(x	PROPN
ejpam-619	166	12	,	,	PUNCT
ejpam-619	166	13	x̄	x̄	PROPN
ejpam-619	166	14	)	)	PUNCT
ejpam-619	166	15	.	.	PUNCT
ejpam-619	167	1	=	=	PUNCT
ejpam-619	167	2	f(x	f(x	PROPN
ejpam-619	167	3	,	,	PUNCT
ejpam-619	167	4	x̄	x̄	NOUN
ejpam-619	167	5	;	;	PUNCT
ejpam-619	167	6	α(x	α(x	PROPN
ejpam-619	167	7	,	,	PUNCT
ejpam-619	167	8	x̄	x̄	PROPN
ejpam-619	167	9	)	)	PUNCT
ejpam-619	167	10	g	g	PROPN
ejpam-619	167	11	(	(	PUNCT
ejpam-619	167	12	x̄	x̄	NOUN
ejpam-619	167	13	)	)	PUNCT
ejpam-619	167	14	g(x	g(x	NOUN
ejpam-619	167	15	)	)	PUNCT
ejpam-619	167	16	{	{	PUNCT
ejpam-619	167	17	∇	∇	X
ejpam-619	167	18	f	f	X
ejpam-619	167	19	(	(	PUNCT
ejpam-619	167	20	x̄	x̄	PROPN
ejpam-619	167	21	)	)	PUNCT
ejpam-619	167	22	g	g	PROPN
ejpam-619	167	23	(	(	PUNCT
ejpam-619	167	24	x̄	x̄	PROPN
ejpam-619	167	25	)	)	PUNCT
ejpam-619	168	1	+	+	CCONJ
ejpam-619	168	2	�	�	PROPN
ejpam-619	168	3	1	1	NUM
ejpam-619	168	4	g	g	PROPN
ejpam-619	168	5	(	(	PUNCT
ejpam-619	168	6	x̄	x̄	PROPN
ejpam-619	168	7	)	)	PUNCT
ejpam-619	169	1	+	+	CCONJ
ejpam-619	169	2	f	f	X
ejpam-619	169	3	(	(	PUNCT
ejpam-619	169	4	x̄	x̄	PROPN
ejpam-619	169	5	)	)	PUNCT
ejpam-619	169	6	g2	g2	PROPN
ejpam-619	169	7	(	(	PUNCT
ejpam-619	169	8	x̄	x̄	PROPN
ejpam-619	169	9	)	)	PUNCT
ejpam-619	169	10	�	�	PROPN
ejpam-619	169	11	∇pk	∇pk	PROPN
ejpam-619	169	12	(	(	PUNCT
ejpam-619	169	13	x̄	x̄	NOUN
ejpam-619	169	14	,	,	PUNCT
ejpam-619	169	15	p	p	NOUN
ejpam-619	169	16	)	)	PUNCT
ejpam-619	169	17	}	}	PUNCT
ejpam-619	169	18	)	)	PUNCT
ejpam-619	170	1	+	+	NOUN
ejpam-619	170	2	β(x	β(x	NOUN
ejpam-619	170	3	,	,	PUNCT
ejpam-619	170	4	x̄	x̄	NUM
ejpam-619	170	5	)	)	PUNCT
ejpam-619	170	6	g	g	PROPN
ejpam-619	170	7	(	(	PUNCT
ejpam-619	170	8	x̄	x̄	NOUN
ejpam-619	170	9	)	)	PUNCT
ejpam-619	170	10	g(x	g(x	NOUN
ejpam-619	170	11	)	)	PUNCT
ejpam-619	170	12	�	�	PROPN
ejpam-619	170	13	1	1	NUM
ejpam-619	170	14	g	g	PROPN
ejpam-619	170	15	(	(	PUNCT
ejpam-619	170	16	x̄	x̄	PROPN
ejpam-619	170	17	)	)	PUNCT
ejpam-619	171	1	+	+	CCONJ
ejpam-619	171	2	f	f	X
ejpam-619	171	3	(	(	PUNCT
ejpam-619	171	4	x̄	x̄	PROPN
ejpam-619	171	5	)	)	PUNCT
ejpam-619	171	6	g2	g2	PROPN
ejpam-619	171	7	(	(	PUNCT
ejpam-619	171	8	x̄	x̄	PROPN
ejpam-619	171	9	)	)	PUNCT
ejpam-619	171	10	�	�	PROPN
ejpam-619	171	11	{	{	PUNCT
ejpam-619	171	12	k	k	X
ejpam-619	171	13	(	(	PUNCT
ejpam-619	171	14	x̄	x̄	NOUN
ejpam-619	171	15	,	,	PUNCT
ejpam-619	171	16	p)−	p)−	PROPN
ejpam-619	171	17	pt∇pk	pt∇pk	PROPN
ejpam-619	171	18	(	(	PUNCT
ejpam-619	171	19	x̄	x̄	NOUN
ejpam-619	171	20	,	,	PUNCT
ejpam-619	171	21	p	p	NOUN
ejpam-619	171	22	)	)	PUNCT
ejpam-619	171	23	}	}	PUNCT
ejpam-619	172	1	+	+	NOUN
ejpam-619	172	2	ρ	ρ	PROPN
ejpam-619	172	3	�	�	PROPN
ejpam-619	172	4	1	1	NUM
ejpam-619	172	5	g(x	g(x	NOUN
ejpam-619	172	6	)	)	PUNCT
ejpam-619	173	1	+	+	CCONJ
ejpam-619	173	2	f	f	X
ejpam-619	173	3	(	(	PUNCT
ejpam-619	173	4	x̄	x̄	PROPN
ejpam-619	173	5	)	)	PUNCT
ejpam-619	173	6	g(x)g	g(x)g	PROPN
ejpam-619	173	7	(	(	PUNCT
ejpam-619	173	8	x̄	x̄	NOUN
ejpam-619	173	9	)	)	PUNCT
ejpam-619	173	10	�	�	PROPN
ejpam-619	173	11	d2(x	d2(x	PROPN
ejpam-619	173	12	,	,	PUNCT
ejpam-619	173	13	x̄	x̄	PROPN
ejpam-619	173	14	)	)	PUNCT
ejpam-619	173	15	.	.	PUNCT
ejpam-619	174	1	therefore	therefore	ADV
ejpam-619	174	2	,	,	PUNCT
ejpam-619	174	3	f	f	PROPN
ejpam-619	174	4	(	(	PUNCT
ejpam-619	174	5	x	x	NOUN
ejpam-619	174	6	)	)	PUNCT
ejpam-619	174	7	g(x	g(x	NOUN
ejpam-619	174	8	)	)	PUNCT
ejpam-619	175	1	−	−	PROPN
ejpam-619	175	2	f	f	PROPN
ejpam-619	175	3	(	(	PUNCT
ejpam-619	175	4	x̄	x̄	PROPN
ejpam-619	175	5	)	)	PUNCT
ejpam-619	175	6	g	g	PROPN
ejpam-619	175	7	(	(	PUNCT
ejpam-619	175	8	x̄	x̄	PROPN
ejpam-619	175	9	)	)	PUNCT
ejpam-619	175	10	≧	≧	X
ejpam-619	176	1	f	f	PROPN
ejpam-619	176	2	�	�	PROPN
ejpam-619	176	3	x	x	SYM
ejpam-619	176	4	,	,	PUNCT
ejpam-619	176	5	x̄	x̄	NOUN
ejpam-619	176	6	;	;	PUNCT
ejpam-619	176	7	ᾱ(x	ᾱ(x	PROPN
ejpam-619	176	8	,	,	PUNCT
ejpam-619	176	9	x̄	x̄	NUM
ejpam-619	176	10	)	)	PUNCT
ejpam-619	176	11	�	�	PROPN
ejpam-619	176	12	∇	∇	X
ejpam-619	176	13	f	f	PROPN
ejpam-619	176	14	(	(	PUNCT
ejpam-619	176	15	x̄	x̄	PROPN
ejpam-619	176	16	)	)	PUNCT
ejpam-619	176	17	g	g	PROPN
ejpam-619	176	18	(	(	PUNCT
ejpam-619	176	19	x̄	x̄	PROPN
ejpam-619	176	20	)	)	PUNCT
ejpam-619	177	1	+	+	PUNCT
ejpam-619	177	2	∇pk̄	∇pk̄	VERB
ejpam-619	177	3	(	(	PUNCT
ejpam-619	177	4	x̄	x̄	NOUN
ejpam-619	177	5	,	,	PUNCT
ejpam-619	177	6	p	p	X
ejpam-619	177	7	)	)	PUNCT
ejpam-619	177	8	�	�	PROPN
ejpam-619	177	9	�	�	PROPN
ejpam-619	177	10	+	+	CCONJ
ejpam-619	177	11	β̄(x	β̄(x	PROPN
ejpam-619	177	12	,	,	PUNCT
ejpam-619	177	13	x̄){k̄	x̄){k̄	PROPN
ejpam-619	177	14	(	(	PUNCT
ejpam-619	177	15	x̄	x̄	NOUN
ejpam-619	177	16	,	,	PUNCT
ejpam-619	177	17	p)−	p)−	NOUN
ejpam-619	177	18	pt∇pk̄	pt∇pk̄	NOUN
ejpam-619	177	19	(	(	PUNCT
ejpam-619	177	20	x̄	x̄	NOUN
ejpam-619	177	21	,	,	PUNCT
ejpam-619	177	22	p)}+ρd̄2(x	p)}+ρd̄2(x	PROPN
ejpam-619	177	23	,	,	PUNCT
ejpam-619	177	24	x̄	x̄	PROPN
ejpam-619	177	25	)	)	PUNCT
ejpam-619	177	26	,	,	PUNCT
ejpam-619	177	27	i.e.	i.e.	X
ejpam-619	177	28	,	,	PUNCT
ejpam-619	177	29	f	f	PROPN
ejpam-619	177	30	(	(	PUNCT
ejpam-619	177	31	x	x	NOUN
ejpam-619	177	32	)	)	PUNCT
ejpam-619	177	33	g(x	g(x	NOUN
ejpam-619	177	34	)	)	PUNCT
ejpam-619	177	35	is	be	AUX
ejpam-619	177	36	higher	high	ADJ
ejpam-619	177	37	-	-	PUNCT
ejpam-619	177	38	order	order	NOUN
ejpam-619	177	39	(	(	PUNCT
ejpam-619	177	40	f	f	NOUN
ejpam-619	177	41	,	,	PUNCT
ejpam-619	177	42	ᾱ	ᾱ	NOUN
ejpam-619	177	43	,	,	PUNCT
ejpam-619	177	44	β̄	β̄	NOUN
ejpam-619	177	45	,	,	PUNCT
ejpam-619	177	46	ρ	ρ	PROPN
ejpam-619	177	47	,	,	PUNCT
ejpam-619	177	48	d̄)-convex	d̄)-convex	PROPN
ejpam-619	177	49	at	at	ADP
ejpam-619	177	50	x̄	x̄	NOUN
ejpam-619	177	51	with	with	ADP
ejpam-619	177	52	respect	respect	NOUN
ejpam-619	177	53	to	to	PART
ejpam-619	177	54	k̄	k̄	VERB
ejpam-619	177	55	.	.	PUNCT
ejpam-619	178	1	in	in	ADP
ejpam-619	178	2	view	view	NOUN
ejpam-619	178	3	of	of	ADP
ejpam-619	178	4	theorem	theorem	NOUN
ejpam-619	178	5	4	4	NUM
ejpam-619	178	6	,	,	PUNCT
ejpam-619	178	7	the	the	DET
ejpam-619	178	8	results	result	NOUN
ejpam-619	178	9	of	of	ADP
ejpam-619	178	10	section	section	NOUN
ejpam-619	178	11	4	4	NUM
ejpam-619	178	12	lead	lead	NOUN
ejpam-619	178	13	to	to	ADP
ejpam-619	178	14	the	the	DET
ejpam-619	178	15	following	follow	VERB
ejpam-619	178	16	duality	duality	NOUN
ejpam-619	178	17	relations	relation	NOUN
ejpam-619	178	18	between	between	ADP
ejpam-619	178	19	(	(	PUNCT
ejpam-619	178	20	fp	fp	NOUN
ejpam-619	178	21	)	)	PUNCT
ejpam-619	178	22	and	and	CCONJ
ejpam-619	178	23	its	its	PRON
ejpam-619	178	24	mond	mond	PROPN
ejpam-619	178	25	-	-	PUNCT
ejpam-619	178	26	weir	weir	NOUN
ejpam-619	178	27	dual	dual	PROPN
ejpam-619	178	28	(	(	PUNCT
ejpam-619	178	29	mfd	mfd	PROPN
ejpam-619	178	30	)	)	PUNCT
ejpam-619	178	31	.	.	PUNCT
ejpam-619	179	1	(	(	PUNCT
ejpam-619	179	2	mfd	mfd	PROPN
ejpam-619	179	3	)	)	PUNCT
ejpam-619	179	4	maximize	maximize	VERB
ejpam-619	179	5	f	f	PROPN
ejpam-619	179	6	(	(	PUNCT
ejpam-619	179	7	u	u	NOUN
ejpam-619	179	8	)	)	PUNCT
ejpam-619	179	9	g(u	g(u	PROPN
ejpam-619	179	10	)	)	PUNCT
ejpam-619	179	11	subject	subject	NOUN
ejpam-619	179	12	to	to	ADP
ejpam-619	179	13	∇	∇	PROPN
ejpam-619	179	14	�	�	PROPN
ejpam-619	179	15	f	f	PROPN
ejpam-619	179	16	(	(	PUNCT
ejpam-619	179	17	u	u	NOUN
ejpam-619	179	18	)	)	PUNCT
ejpam-619	179	19	g(u	g(u	PROPN
ejpam-619	179	20	)	)	PUNCT
ejpam-619	179	21	�	�	PROPN
ejpam-619	180	1	+	+	NOUN
ejpam-619	180	2	∇h(u)v	∇h(u)v	PROPN
ejpam-619	180	3	=	=	SYM
ejpam-619	180	4	0	0	PROPN
ejpam-619	180	5	,	,	PUNCT
ejpam-619	180	6	vt	vt	PROPN
ejpam-619	180	7	h(u)≧	h(u)≧	PROPN
ejpam-619	180	8	0	0	NUM
ejpam-619	180	9	,	,	PUNCT
ejpam-619	180	10	u	u	NOUN
ejpam-619	181	1	∈	∈	PROPN
ejpam-619	181	2	x	x	X
ejpam-619	181	3	,	,	PUNCT
ejpam-619	181	4	v	v	PART
ejpam-619	181	5	≧	≧	NOUN
ejpam-619	181	6	0	0	NUM
ejpam-619	181	7	,	,	PUNCT
ejpam-619	181	8	v	v	PROPN
ejpam-619	181	9	∈	∈	PROPN
ejpam-619	181	10	rm	rm	NOUN
ejpam-619	181	11	.	.	PUNCT
ejpam-619	182	1	t.	t.	PROPN
ejpam-619	182	2	gulati	gulati	PROPN
ejpam-619	182	3	,	,	PUNCT
ejpam-619	182	4	h.	h.	PROPN
ejpam-619	182	5	saini	saini	PROPN
ejpam-619	182	6	/	/	SYM
ejpam-619	182	7	eur	eur	PROPN
ejpam-619	182	8	.	.	PUNCT
ejpam-619	183	1	j.	j.	PROPN
ejpam-619	183	2	pure	pure	PROPN
ejpam-619	183	3	appl	appl	PROPN
ejpam-619	183	4	.	.	PROPN
ejpam-619	183	5	math	math	PROPN
ejpam-619	183	6	,	,	PUNCT
ejpam-619	183	7	4	4	NUM
ejpam-619	183	8	(	(	PUNCT
ejpam-619	183	9	2011	2011	NUM
ejpam-619	183	10	)	)	PUNCT
ejpam-619	183	11	,	,	PUNCT
ejpam-619	183	12	266	266	NUM
ejpam-619	183	13	-	-	SYM
ejpam-619	183	14	275	275	NUM
ejpam-619	183	15	273	273	NUM
ejpam-619	183	16	theorem	theorem	NOUN
ejpam-619	183	17	5	5	NUM
ejpam-619	183	18	(	(	PUNCT
ejpam-619	183	19	weak	weak	ADJ
ejpam-619	183	20	duality	duality	NOUN
ejpam-619	183	21	)	)	PUNCT
ejpam-619	183	22	.	.	PUNCT
ejpam-619	184	1	let	let	VERB
ejpam-619	184	2	x	x	PRON
ejpam-619	184	3	and	and	CCONJ
ejpam-619	184	4	(	(	PUNCT
ejpam-619	184	5	u	u	NOUN
ejpam-619	184	6	,	,	PUNCT
ejpam-619	184	7	v	v	NOUN
ejpam-619	184	8	)	)	PUNCT
ejpam-619	184	9	be	be	AUX
ejpam-619	184	10	feasible	feasible	ADJ
ejpam-619	184	11	solutions	solution	NOUN
ejpam-619	184	12	of	of	ADP
ejpam-619	184	13	(	(	PUNCT
ejpam-619	184	14	fp	fp	NOUN
ejpam-619	184	15	)	)	PUNCT
ejpam-619	184	16	and	and	CCONJ
ejpam-619	184	17	(	(	PUNCT
ejpam-619	184	18	mfd	mfd	PROPN
ejpam-619	184	19	)	)	PUNCT
ejpam-619	184	20	respectively	respectively	ADV
ejpam-619	184	21	.	.	PUNCT
ejpam-619	185	1	let	let	VERB
ejpam-619	185	2	(	(	PUNCT
ejpam-619	185	3	i	i	NOUN
ejpam-619	185	4	)	)	PUNCT
ejpam-619	185	5	f	f	PROPN
ejpam-619	185	6	and	and	CCONJ
ejpam-619	185	7	−g	−g	NOUN
ejpam-619	185	8	be	be	AUX
ejpam-619	185	9	higher	high	ADJ
ejpam-619	185	10	-	-	PUNCT
ejpam-619	185	11	order	order	NOUN
ejpam-619	185	12	(	(	PUNCT
ejpam-619	185	13	f	f	X
ejpam-619	185	14	,	,	PUNCT
ejpam-619	185	15	α	α	PROPN
ejpam-619	185	16	,	,	PUNCT
ejpam-619	185	17	β	β	X
ejpam-619	185	18	,	,	PUNCT
ejpam-619	185	19	ρ1	ρ1	NOUN
ejpam-619	185	20	,	,	PUNCT
ejpam-619	185	21	d)-convex	d)-convex	PUNCT
ejpam-619	185	22	at	at	ADP
ejpam-619	185	23	u	u	NOUN
ejpam-619	185	24	with	with	ADP
ejpam-619	185	25	respect	respect	NOUN
ejpam-619	185	26	to	to	ADP
ejpam-619	185	27	k	k	PROPN
ejpam-619	185	28	,	,	PUNCT
ejpam-619	185	29	(	(	PUNCT
ejpam-619	185	30	ii	ii	NOUN
ejpam-619	185	31	)	)	PUNCT
ejpam-619	185	32	vt	vt	PROPN
ejpam-619	185	33	h	h	PROPN
ejpam-619	185	34	be	be	AUX
ejpam-619	185	35	higher	high	ADJ
ejpam-619	185	36	-	-	PUNCT
ejpam-619	185	37	order	order	NOUN
ejpam-619	185	38	(	(	PUNCT
ejpam-619	185	39	f	f	NOUN
ejpam-619	185	40	,	,	PUNCT
ejpam-619	185	41	ᾱ	ᾱ	NOUN
ejpam-619	185	42	,	,	PUNCT
ejpam-619	185	43	β̄	β̄	ADJ
ejpam-619	185	44	,	,	PUNCT
ejpam-619	185	45	ρ2	ρ2	NOUN
ejpam-619	185	46	,	,	PUNCT
ejpam-619	185	47	d̄)-convex	d̄)-convex	PROPN
ejpam-619	185	48	at	at	ADP
ejpam-619	185	49	u	u	NOUN
ejpam-619	185	50	with	with	ADP
ejpam-619	185	51	respect	respect	NOUN
ejpam-619	185	52	to	to	ADP
ejpam-619	185	53	−k̄	−k̄	NOUN
ejpam-619	185	54	,	,	PUNCT
ejpam-619	185	55	where	where	SCONJ
ejpam-619	185	56	ᾱ	ᾱ	NOUN
ejpam-619	185	57	,	,	PUNCT
ejpam-619	185	58	β̄	β̄	ADJ
ejpam-619	185	59	,	,	PUNCT
ejpam-619	185	60	k̄	k̄	VERB
ejpam-619	185	61	and	and	CCONJ
ejpam-619	185	62	d̄	d̄	NOUN
ejpam-619	185	63	are	be	AUX
ejpam-619	185	64	as	as	SCONJ
ejpam-619	185	65	given	give	VERB
ejpam-619	185	66	in	in	ADP
ejpam-619	185	67	theorem	theorem	ADJ
ejpam-619	185	68	4	4	NUM
ejpam-619	185	69	,	,	PUNCT
ejpam-619	185	70	and	and	CCONJ
ejpam-619	185	71	(	(	PUNCT
ejpam-619	185	72	iii	iii	NOUN
ejpam-619	185	73	)	)	PUNCT
ejpam-619	185	74	ρ1	ρ1	NOUN
ejpam-619	185	75	+	+	NOUN
ejpam-619	185	76	ρ2	ρ2	NOUN
ejpam-619	185	77	≧	≧	NOUN
ejpam-619	185	78	0	0	X
ejpam-619	185	79	.	.	PUNCT
ejpam-619	186	1	then	then	ADV
ejpam-619	186	2	f	f	X
ejpam-619	186	3	(	(	PUNCT
ejpam-619	186	4	x	x	NOUN
ejpam-619	186	5	)	)	PUNCT
ejpam-619	186	6	g(x	g(x	NOUN
ejpam-619	186	7	)	)	PUNCT
ejpam-619	186	8	≧	≧	X
ejpam-619	187	1	f	f	X
ejpam-619	187	2	(	(	PUNCT
ejpam-619	187	3	u	u	NOUN
ejpam-619	187	4	)	)	PUNCT
ejpam-619	187	5	g(u	g(u	PROPN
ejpam-619	187	6	)	)	PUNCT
ejpam-619	187	7	.	.	PUNCT
ejpam-619	188	1	theorem	theorem	ADJ
ejpam-619	188	2	6	6	NUM
ejpam-619	188	3	(	(	PUNCT
ejpam-619	188	4	strong	strong	ADJ
ejpam-619	188	5	duality	duality	NOUN
ejpam-619	188	6	)	)	PUNCT
ejpam-619	188	7	.	.	PUNCT
ejpam-619	189	1	let	let	VERB
ejpam-619	189	2	x̄	x̄	PRON
ejpam-619	189	3	be	be	AUX
ejpam-619	189	4	an	an	DET
ejpam-619	189	5	optimal	optimal	ADJ
ejpam-619	189	6	solution	solution	NOUN
ejpam-619	189	7	of	of	ADP
ejpam-619	189	8	the	the	DET
ejpam-619	189	9	problem	problem	NOUN
ejpam-619	189	10	(	(	PUNCT
ejpam-619	189	11	fp	fp	X
ejpam-619	189	12	)	)	PUNCT
ejpam-619	189	13	and	and	CCONJ
ejpam-619	189	14	let	let	VERB
ejpam-619	189	15	h	h	PRON
ejpam-619	189	16	satisfy	satisfy	VERB
ejpam-619	189	17	a	a	DET
ejpam-619	189	18	constraint	constraint	NOUN
ejpam-619	189	19	qualification	qualification	NOUN
ejpam-619	189	20	.	.	PUNCT
ejpam-619	190	1	further	far	ADV
ejpam-619	190	2	,	,	PUNCT
ejpam-619	190	3	let	let	VERB
ejpam-619	190	4	theorem	theorem	VERB
ejpam-619	190	5	5	5	NUM
ejpam-619	190	6	hold	hold	VERB
ejpam-619	190	7	for	for	ADP
ejpam-619	190	8	the	the	DET
ejpam-619	190	9	feasible	feasible	ADJ
ejpam-619	190	10	solution	solution	NOUN
ejpam-619	190	11	x̄	x̄	PROPN
ejpam-619	190	12	of	of	ADP
ejpam-619	190	13	(	(	PUNCT
ejpam-619	190	14	fp	fp	NOUN
ejpam-619	190	15	)	)	PUNCT
ejpam-619	190	16	and	and	CCONJ
ejpam-619	190	17	all	all	DET
ejpam-619	190	18	feasible	feasible	ADJ
ejpam-619	190	19	solutions	solution	NOUN
ejpam-619	190	20	(	(	PUNCT
ejpam-619	190	21	u	u	NOUN
ejpam-619	190	22	,	,	PUNCT
ejpam-619	190	23	v	v	NOUN
ejpam-619	190	24	)	)	PUNCT
ejpam-619	190	25	of	of	ADP
ejpam-619	190	26	(	(	PUNCT
ejpam-619	190	27	mfd	mfd	PROPN
ejpam-619	190	28	)	)	PUNCT
ejpam-619	190	29	.	.	PUNCT
ejpam-619	191	1	then	then	ADV
ejpam-619	191	2	there	there	PRON
ejpam-619	191	3	exists	exist	VERB
ejpam-619	191	4	a	a	DET
ejpam-619	191	5	v̄	v̄	NOUN
ejpam-619	191	6	∈	∈	PROPN
ejpam-619	191	7	rm	rm	NOUN
ejpam-619	191	8	+	+	CCONJ
ejpam-619	191	9	such	such	ADJ
ejpam-619	191	10	that	that	SCONJ
ejpam-619	191	11	(	(	PUNCT
ejpam-619	191	12	x̄	x̄	NOUN
ejpam-619	191	13	,	,	PUNCT
ejpam-619	191	14	v̄	v̄	PROPN
ejpam-619	191	15	)	)	PUNCT
ejpam-619	191	16	is	be	AUX
ejpam-619	191	17	an	an	DET
ejpam-619	191	18	optimal	optimal	ADJ
ejpam-619	191	19	solution	solution	NOUN
ejpam-619	191	20	of	of	ADP
ejpam-619	191	21	(	(	PUNCT
ejpam-619	191	22	mfd	mfd	PROPN
ejpam-619	191	23	)	)	PUNCT
ejpam-619	191	24	.	.	PUNCT
ejpam-619	192	1	6	6	X
ejpam-619	192	2	.	.	X
ejpam-619	192	3	wolfe	wolfe	PROPN
ejpam-619	192	4	duality	duality	PROPN
ejpam-619	192	5	the	the	DET
ejpam-619	192	6	wolfe	wolfe	PROPN
ejpam-619	192	7	dual	dual	ADJ
ejpam-619	192	8	of	of	ADP
ejpam-619	192	9	(	(	PUNCT
ejpam-619	192	10	np	np	INTJ
ejpam-619	192	11	)	)	PUNCT
ejpam-619	192	12	and	and	CCONJ
ejpam-619	192	13	(	(	PUNCT
ejpam-619	192	14	fp	fp	X
ejpam-619	192	15	)	)	PUNCT
ejpam-619	192	16	are	be	AUX
ejpam-619	192	17	respectively	respectively	ADV
ejpam-619	192	18	(	(	PUNCT
ejpam-619	192	19	wd	wd	ADJ
ejpam-619	192	20	)	)	PUNCT
ejpam-619	192	21	maximize	maximize	VERB
ejpam-619	192	22	φ(u	φ(u	NOUN
ejpam-619	192	23	)	)	PUNCT
ejpam-619	193	1	+	+	NUM
ejpam-619	193	2	vt	vt	PROPN
ejpam-619	193	3	h(u	h(u	PROPN
ejpam-619	193	4	)	)	PUNCT
ejpam-619	193	5	subject	subject	ADJ
ejpam-619	193	6	to	to	ADP
ejpam-619	193	7	∇φ(u	∇φ(u	NOUN
ejpam-619	193	8	)	)	PUNCT
ejpam-619	194	1	+	+	NOUN
ejpam-619	194	2	∇h(u)v	∇h(u)v	NOUN
ejpam-619	194	3	=	=	SYM
ejpam-619	194	4	0	0	NUM
ejpam-619	194	5	,	,	PUNCT
ejpam-619	194	6	u	u	NOUN
ejpam-619	194	7	∈	∈	PROPN
ejpam-619	194	8	x	x	X
ejpam-619	194	9	,	,	PUNCT
ejpam-619	194	10	v	v	PART
ejpam-619	194	11	≧	≧	NOUN
ejpam-619	194	12	0	0	NUM
ejpam-619	194	13	,	,	PUNCT
ejpam-619	194	14	v	v	ADP
ejpam-619	194	15	∈	∈	PROPN
ejpam-619	194	16	rm	rm	NOUN
ejpam-619	194	17	,	,	PUNCT
ejpam-619	194	18	(	(	PUNCT
ejpam-619	194	19	wfd	wfd	NOUN
ejpam-619	194	20	)	)	PUNCT
ejpam-619	194	21	maximize	maximize	VERB
ejpam-619	194	22	f	f	PROPN
ejpam-619	194	23	(	(	PUNCT
ejpam-619	194	24	u	u	NOUN
ejpam-619	194	25	)	)	PUNCT
ejpam-619	194	26	g(u	g(u	PROPN
ejpam-619	194	27	)	)	PUNCT
ejpam-619	195	1	+	+	NUM
ejpam-619	195	2	vt	vt	PROPN
ejpam-619	195	3	h(u	h(u	PROPN
ejpam-619	195	4	)	)	PUNCT
ejpam-619	195	5	subject	subject	ADJ
ejpam-619	195	6	to	to	ADP
ejpam-619	195	7	∇	∇	PROPN
ejpam-619	195	8	�	�	PROPN
ejpam-619	195	9	f	f	PROPN
ejpam-619	195	10	(	(	PUNCT
ejpam-619	195	11	u	u	NOUN
ejpam-619	195	12	)	)	PUNCT
ejpam-619	195	13	g(u	g(u	PROPN
ejpam-619	195	14	)	)	PUNCT
ejpam-619	195	15	�	�	PROPN
ejpam-619	196	1	+	+	NOUN
ejpam-619	196	2	∇h(u)v	∇h(u)v	PROPN
ejpam-619	196	3	=	=	SYM
ejpam-619	196	4	0	0	NUM
ejpam-619	196	5	,	,	PUNCT
ejpam-619	196	6	u	u	NOUN
ejpam-619	196	7	∈	∈	PROPN
ejpam-619	196	8	x	x	X
ejpam-619	196	9	,	,	PUNCT
ejpam-619	196	10	v	v	PART
ejpam-619	196	11	≧	≧	NOUN
ejpam-619	196	12	0	0	NUM
ejpam-619	196	13	,	,	PUNCT
ejpam-619	196	14	v	v	PROPN
ejpam-619	196	15	∈	∈	PROPN
ejpam-619	196	16	rm	rm	NOUN
ejpam-619	196	17	.	.	PUNCT
ejpam-619	197	1	now	now	ADV
ejpam-619	197	2	we	we	PRON
ejpam-619	197	3	state	state	VERB
ejpam-619	197	4	duality	duality	NOUN
ejpam-619	197	5	relations	relation	NOUN
ejpam-619	197	6	for	for	ADP
ejpam-619	197	7	the	the	DET
ejpam-619	197	8	primal	primal	ADJ
ejpam-619	197	9	problems	problem	NOUN
ejpam-619	197	10	(	(	PUNCT
ejpam-619	197	11	np	np	INTJ
ejpam-619	197	12	)	)	PUNCT
ejpam-619	197	13	and	and	CCONJ
ejpam-619	197	14	(	(	PUNCT
ejpam-619	197	15	fp	fp	X
ejpam-619	197	16	)	)	PUNCT
ejpam-619	197	17	and	and	CCONJ
ejpam-619	197	18	their	their	PRON
ejpam-619	197	19	wolfe	wolfe	PROPN
ejpam-619	197	20	duals	dual	NOUN
ejpam-619	197	21	(	(	PUNCT
ejpam-619	197	22	wd	wd	PROPN
ejpam-619	197	23	)	)	PUNCT
ejpam-619	197	24	and	and	CCONJ
ejpam-619	197	25	(	(	PUNCT
ejpam-619	197	26	wfd	wfd	NOUN
ejpam-619	197	27	)	)	PUNCT
ejpam-619	197	28	respectively	respectively	ADV
ejpam-619	197	29	.	.	PUNCT
ejpam-619	198	1	their	their	PRON
ejpam-619	198	2	proofs	proof	NOUN
ejpam-619	198	3	follow	follow	VERB
ejpam-619	198	4	as	as	ADP
ejpam-619	198	5	in	in	ADP
ejpam-619	198	6	section	section	NOUN
ejpam-619	198	7	4	4	NUM
ejpam-619	198	8	.	.	PUNCT
ejpam-619	199	1	theorem	theorem	VERB
ejpam-619	199	2	7	7	NUM
ejpam-619	199	3	(	(	PUNCT
ejpam-619	199	4	weak	weak	ADJ
ejpam-619	199	5	duality	duality	NOUN
ejpam-619	199	6	)	)	PUNCT
ejpam-619	199	7	.	.	PUNCT
ejpam-619	200	1	let	let	VERB
ejpam-619	200	2	x	x	PRON
ejpam-619	200	3	and	and	CCONJ
ejpam-619	200	4	(	(	PUNCT
ejpam-619	200	5	u	u	NOUN
ejpam-619	200	6	,	,	PUNCT
ejpam-619	200	7	v	v	NOUN
ejpam-619	200	8	)	)	PUNCT
ejpam-619	200	9	be	be	AUX
ejpam-619	200	10	feasible	feasible	ADJ
ejpam-619	200	11	solutions	solution	NOUN
ejpam-619	200	12	of	of	ADP
ejpam-619	200	13	(	(	PUNCT
ejpam-619	200	14	np	np	INTJ
ejpam-619	200	15	)	)	PUNCT
ejpam-619	200	16	and	and	CCONJ
ejpam-619	200	17	(	(	PUNCT
ejpam-619	200	18	wd	wd	PROPN
ejpam-619	200	19	)	)	PUNCT
ejpam-619	200	20	respectively	respectively	ADV
ejpam-619	200	21	.	.	PUNCT
ejpam-619	201	1	let	let	VERB
ejpam-619	201	2	(	(	PUNCT
ejpam-619	201	3	i	i	NOUN
ejpam-619	201	4	)	)	PUNCT
ejpam-619	201	5	φ	φ	PROPN
ejpam-619	201	6	be	be	VERB
ejpam-619	201	7	higher	high	ADJ
ejpam-619	201	8	-	-	PUNCT
ejpam-619	201	9	order	order	NOUN
ejpam-619	201	10	(	(	PUNCT
ejpam-619	201	11	f	f	X
ejpam-619	201	12	,	,	PUNCT
ejpam-619	201	13	α	α	PROPN
ejpam-619	201	14	,	,	PUNCT
ejpam-619	201	15	β	β	X
ejpam-619	201	16	,	,	PUNCT
ejpam-619	201	17	ρ1	ρ1	NOUN
ejpam-619	201	18	,	,	PUNCT
ejpam-619	201	19	d)-convex	d)-convex	PUNCT
ejpam-619	201	20	at	at	ADP
ejpam-619	201	21	u	u	NOUN
ejpam-619	201	22	with	with	ADP
ejpam-619	201	23	respect	respect	NOUN
ejpam-619	201	24	to	to	ADP
ejpam-619	201	25	k	k	PROPN
ejpam-619	201	26	,	,	PUNCT
ejpam-619	201	27	(	(	PUNCT
ejpam-619	201	28	ii	ii	NOUN
ejpam-619	201	29	)	)	PUNCT
ejpam-619	201	30	vt	vt	PROPN
ejpam-619	201	31	h	h	PROPN
ejpam-619	201	32	be	be	AUX
ejpam-619	201	33	higher	high	ADJ
ejpam-619	201	34	-	-	PUNCT
ejpam-619	201	35	order	order	NOUN
ejpam-619	201	36	(	(	PUNCT
ejpam-619	201	37	f	f	X
ejpam-619	201	38	,	,	PUNCT
ejpam-619	201	39	α	α	PROPN
ejpam-619	201	40	,	,	PUNCT
ejpam-619	201	41	β	β	X
ejpam-619	201	42	,	,	PUNCT
ejpam-619	201	43	ρ2	ρ2	NOUN
ejpam-619	201	44	,	,	PUNCT
ejpam-619	201	45	d)-convex	d)-convex	NOUN
ejpam-619	201	46	at	at	ADP
ejpam-619	201	47	u	u	NOUN
ejpam-619	201	48	with	with	ADP
ejpam-619	201	49	respect	respect	NOUN
ejpam-619	201	50	to	to	ADP
ejpam-619	201	51	−k	−k	ADJ
ejpam-619	201	52	,	,	PUNCT
ejpam-619	201	53	and	and	CCONJ
ejpam-619	201	54	(	(	PUNCT
ejpam-619	201	55	iii	iii	NOUN
ejpam-619	201	56	)	)	PUNCT
ejpam-619	201	57	ρ1	ρ1	NOUN
ejpam-619	201	58	+	+	NOUN
ejpam-619	201	59	ρ2	ρ2	NOUN
ejpam-619	201	60	≧	≧	NOUN
ejpam-619	201	61	0	0	X
ejpam-619	201	62	.	.	PUNCT
ejpam-619	202	1	then	then	ADV
ejpam-619	202	2	φ(x)≧	φ(x)≧	PROPN
ejpam-619	202	3	φ(u	φ(u	NOUN
ejpam-619	202	4	)	)	PUNCT
ejpam-619	203	1	+	+	NUM
ejpam-619	203	2	vt	vt	PROPN
ejpam-619	203	3	h(u	h(u	PROPN
ejpam-619	203	4	)	)	PUNCT
ejpam-619	203	5	.	.	PUNCT
ejpam-619	204	1	references	reference	NOUN
ejpam-619	204	2	274	274	NUM
ejpam-619	204	3	theorem	theorem	NOUN
ejpam-619	204	4	8	8	NUM
ejpam-619	204	5	(	(	PUNCT
ejpam-619	204	6	strong	strong	ADJ
ejpam-619	204	7	duality	duality	NOUN
ejpam-619	204	8	)	)	PUNCT
ejpam-619	204	9	.	.	PUNCT
ejpam-619	205	1	let	let	VERB
ejpam-619	205	2	x̄	x̄	PRON
ejpam-619	205	3	be	be	AUX
ejpam-619	205	4	an	an	DET
ejpam-619	205	5	optimal	optimal	ADJ
ejpam-619	205	6	solution	solution	NOUN
ejpam-619	205	7	of	of	ADP
ejpam-619	205	8	the	the	DET
ejpam-619	205	9	problem	problem	NOUN
ejpam-619	205	10	(	(	PUNCT
ejpam-619	205	11	np	np	INTJ
ejpam-619	205	12	)	)	PUNCT
ejpam-619	205	13	and	and	CCONJ
ejpam-619	205	14	let	let	VERB
ejpam-619	205	15	h	h	PRON
ejpam-619	205	16	satisfy	satisfy	VERB
ejpam-619	205	17	a	a	DET
ejpam-619	205	18	constraint	constraint	NOUN
ejpam-619	205	19	qualification	qualification	NOUN
ejpam-619	205	20	.	.	PUNCT
ejpam-619	206	1	further	far	ADV
ejpam-619	206	2	,	,	PUNCT
ejpam-619	206	3	let	let	VERB
ejpam-619	206	4	theorem	theorem	VERB
ejpam-619	206	5	7	7	NUM
ejpam-619	206	6	hold	hold	NOUN
ejpam-619	206	7	for	for	ADP
ejpam-619	206	8	the	the	DET
ejpam-619	206	9	feasible	feasible	ADJ
ejpam-619	206	10	solution	solution	NOUN
ejpam-619	206	11	x̄	x̄	PROPN
ejpam-619	206	12	of	of	ADP
ejpam-619	206	13	(	(	PUNCT
ejpam-619	206	14	np	np	INTJ
ejpam-619	206	15	)	)	PUNCT
ejpam-619	206	16	and	and	CCONJ
ejpam-619	206	17	all	all	DET
ejpam-619	206	18	feasible	feasible	ADJ
ejpam-619	206	19	solutions	solution	NOUN
ejpam-619	206	20	(	(	PUNCT
ejpam-619	206	21	u	u	NOUN
ejpam-619	206	22	,	,	PUNCT
ejpam-619	206	23	v	v	NOUN
ejpam-619	206	24	)	)	PUNCT
ejpam-619	206	25	of	of	ADP
ejpam-619	206	26	(	(	PUNCT
ejpam-619	206	27	wd	wd	PROPN
ejpam-619	206	28	)	)	PUNCT
ejpam-619	206	29	.	.	PUNCT
ejpam-619	207	1	then	then	ADV
ejpam-619	207	2	there	there	PRON
ejpam-619	207	3	exists	exist	VERB
ejpam-619	207	4	a	a	DET
ejpam-619	207	5	v̄	v̄	NOUN
ejpam-619	207	6	∈	∈	PROPN
ejpam-619	207	7	rm	rm	NOUN
ejpam-619	207	8	+	+	CCONJ
ejpam-619	207	9	such	such	ADJ
ejpam-619	207	10	that	that	SCONJ
ejpam-619	207	11	(	(	PUNCT
ejpam-619	207	12	x̄	x̄	NOUN
ejpam-619	207	13	,	,	PUNCT
ejpam-619	207	14	v̄	v̄	PROPN
ejpam-619	207	15	)	)	PUNCT
ejpam-619	207	16	is	be	AUX
ejpam-619	207	17	an	an	DET
ejpam-619	207	18	optimal	optimal	ADJ
ejpam-619	207	19	solution	solution	NOUN
ejpam-619	207	20	of	of	ADP
ejpam-619	207	21	(	(	PUNCT
ejpam-619	207	22	wd	wd	PROPN
ejpam-619	207	23	)	)	PUNCT
ejpam-619	207	24	and	and	CCONJ
ejpam-619	207	25	the	the	DET
ejpam-619	207	26	optimal	optimal	ADJ
ejpam-619	207	27	objective	objective	ADJ
ejpam-619	207	28	function	function	NOUN
ejpam-619	207	29	values	value	NOUN
ejpam-619	207	30	of	of	ADP
ejpam-619	207	31	(	(	PUNCT
ejpam-619	207	32	np	np	INTJ
ejpam-619	207	33	)	)	PUNCT
ejpam-619	207	34	and	and	CCONJ
ejpam-619	207	35	(	(	PUNCT
ejpam-619	207	36	wd	wd	X
ejpam-619	207	37	)	)	PUNCT
ejpam-619	207	38	are	be	AUX
ejpam-619	207	39	equal	equal	ADJ
ejpam-619	207	40	.	.	PUNCT
ejpam-619	208	1	theorem	theorem	ADJ
ejpam-619	208	2	9	9	NUM
ejpam-619	208	3	(	(	PUNCT
ejpam-619	208	4	weak	weak	ADJ
ejpam-619	208	5	duality	duality	NOUN
ejpam-619	208	6	)	)	PUNCT
ejpam-619	208	7	.	.	PUNCT
ejpam-619	209	1	let	let	VERB
ejpam-619	209	2	x	x	PRON
ejpam-619	209	3	and	and	CCONJ
ejpam-619	209	4	(	(	PUNCT
ejpam-619	209	5	u	u	NOUN
ejpam-619	209	6	,	,	PUNCT
ejpam-619	209	7	v	v	NOUN
ejpam-619	209	8	)	)	PUNCT
ejpam-619	209	9	be	be	AUX
ejpam-619	209	10	feasible	feasible	ADJ
ejpam-619	209	11	solutions	solution	NOUN
ejpam-619	209	12	of	of	ADP
ejpam-619	209	13	(	(	PUNCT
ejpam-619	209	14	fp	fp	NOUN
ejpam-619	209	15	)	)	PUNCT
ejpam-619	209	16	and	and	CCONJ
ejpam-619	209	17	(	(	PUNCT
ejpam-619	209	18	wfd	wfd	NOUN
ejpam-619	209	19	)	)	PUNCT
ejpam-619	209	20	respectively	respectively	ADV
ejpam-619	209	21	.	.	PUNCT
ejpam-619	210	1	let	let	VERB
ejpam-619	210	2	(	(	PUNCT
ejpam-619	210	3	i	i	NOUN
ejpam-619	210	4	)	)	PUNCT
ejpam-619	210	5	f	f	PROPN
ejpam-619	210	6	and	and	CCONJ
ejpam-619	210	7	−g	−g	NOUN
ejpam-619	210	8	be	be	AUX
ejpam-619	210	9	higher	high	ADJ
ejpam-619	210	10	-	-	PUNCT
ejpam-619	210	11	order	order	NOUN
ejpam-619	210	12	(	(	PUNCT
ejpam-619	210	13	f	f	X
ejpam-619	210	14	,	,	PUNCT
ejpam-619	210	15	α	α	PROPN
ejpam-619	210	16	,	,	PUNCT
ejpam-619	210	17	β	β	X
ejpam-619	210	18	,	,	PUNCT
ejpam-619	210	19	ρ1	ρ1	NOUN
ejpam-619	210	20	,	,	PUNCT
ejpam-619	210	21	d)-convex	d)-convex	PUNCT
ejpam-619	210	22	at	at	ADP
ejpam-619	210	23	u	u	NOUN
ejpam-619	210	24	with	with	ADP
ejpam-619	210	25	respect	respect	NOUN
ejpam-619	210	26	to	to	ADP
ejpam-619	210	27	k	k	PROPN
ejpam-619	210	28	,	,	PUNCT
ejpam-619	210	29	(	(	PUNCT
ejpam-619	210	30	ii	ii	NOUN
ejpam-619	210	31	)	)	PUNCT
ejpam-619	210	32	vt	vt	PROPN
ejpam-619	210	33	h	h	PROPN
ejpam-619	210	34	be	be	AUX
ejpam-619	210	35	higher	high	ADJ
ejpam-619	210	36	-	-	PUNCT
ejpam-619	210	37	order	order	NOUN
ejpam-619	210	38	(	(	PUNCT
ejpam-619	210	39	f	f	NOUN
ejpam-619	210	40	,	,	PUNCT
ejpam-619	210	41	ᾱ	ᾱ	NOUN
ejpam-619	210	42	,	,	PUNCT
ejpam-619	210	43	β̄	β̄	ADJ
ejpam-619	210	44	,	,	PUNCT
ejpam-619	210	45	ρ2	ρ2	NOUN
ejpam-619	210	46	,	,	PUNCT
ejpam-619	210	47	d̄)-convex	d̄)-convex	PROPN
ejpam-619	210	48	at	at	ADP
ejpam-619	210	49	u	u	NOUN
ejpam-619	210	50	with	with	ADP
ejpam-619	210	51	respect	respect	NOUN
ejpam-619	210	52	to	to	ADP
ejpam-619	210	53	−k̄	−k̄	NOUN
ejpam-619	210	54	,	,	PUNCT
ejpam-619	210	55	where	where	SCONJ
ejpam-619	210	56	ᾱ	ᾱ	NOUN
ejpam-619	210	57	,	,	PUNCT
ejpam-619	210	58	β̄	β̄	ADJ
ejpam-619	210	59	,	,	PUNCT
ejpam-619	210	60	k̄	k̄	VERB
ejpam-619	210	61	and	and	CCONJ
ejpam-619	210	62	d̄	d̄	NOUN
ejpam-619	210	63	are	be	AUX
ejpam-619	210	64	as	as	SCONJ
ejpam-619	210	65	given	give	VERB
ejpam-619	210	66	in	in	ADP
ejpam-619	210	67	theorem	theorem	ADJ
ejpam-619	210	68	4	4	NUM
ejpam-619	210	69	,	,	PUNCT
ejpam-619	210	70	and	and	CCONJ
ejpam-619	210	71	(	(	PUNCT
ejpam-619	210	72	iii	iii	NOUN
ejpam-619	210	73	)	)	PUNCT
ejpam-619	210	74	ρ1	ρ1	NOUN
ejpam-619	210	75	+	+	NOUN
ejpam-619	210	76	ρ2	ρ2	NOUN
ejpam-619	210	77	≧	≧	NOUN
ejpam-619	210	78	0	0	X
ejpam-619	210	79	.	.	PUNCT
ejpam-619	211	1	then	then	ADV
ejpam-619	211	2	f	f	X
ejpam-619	211	3	(	(	PUNCT
ejpam-619	211	4	x	x	NOUN
ejpam-619	211	5	)	)	PUNCT
ejpam-619	211	6	g(x	g(x	NOUN
ejpam-619	211	7	)	)	PUNCT
ejpam-619	211	8	≧	≧	X
ejpam-619	212	1	f	f	X
ejpam-619	212	2	(	(	PUNCT
ejpam-619	212	3	u	u	NOUN
ejpam-619	212	4	)	)	PUNCT
ejpam-619	212	5	g(u	g(u	PROPN
ejpam-619	212	6	)	)	PUNCT
ejpam-619	213	1	+	+	NUM
ejpam-619	213	2	vt	vt	PROPN
ejpam-619	213	3	h(u	h(u	PROPN
ejpam-619	213	4	)	)	PUNCT
ejpam-619	213	5	.	.	PUNCT
ejpam-619	214	1	theorem	theorem	ADJ
ejpam-619	214	2	10	10	NUM
ejpam-619	214	3	(	(	PUNCT
ejpam-619	214	4	strong	strong	ADJ
ejpam-619	214	5	duality	duality	NOUN
ejpam-619	214	6	)	)	PUNCT
ejpam-619	214	7	.	.	PUNCT
ejpam-619	215	1	let	let	VERB
ejpam-619	215	2	x̄	x̄	PRON
ejpam-619	215	3	be	be	AUX
ejpam-619	215	4	an	an	DET
ejpam-619	215	5	optimal	optimal	ADJ
ejpam-619	215	6	solution	solution	NOUN
ejpam-619	215	7	of	of	ADP
ejpam-619	215	8	the	the	DET
ejpam-619	215	9	problem	problem	NOUN
ejpam-619	215	10	(	(	PUNCT
ejpam-619	215	11	fp	fp	X
ejpam-619	215	12	)	)	PUNCT
ejpam-619	215	13	and	and	CCONJ
ejpam-619	215	14	let	let	VERB
ejpam-619	215	15	h	h	PRON
ejpam-619	215	16	satisfy	satisfy	VERB
ejpam-619	215	17	a	a	DET
ejpam-619	215	18	constraint	constraint	NOUN
ejpam-619	215	19	qualification	qualification	NOUN
ejpam-619	215	20	.	.	PUNCT
ejpam-619	216	1	further	far	ADV
ejpam-619	216	2	,	,	PUNCT
ejpam-619	216	3	let	let	VERB
ejpam-619	216	4	theorem	theorem	VERB
ejpam-619	216	5	9	9	NUM
ejpam-619	216	6	hold	hold	NOUN
ejpam-619	216	7	for	for	ADP
ejpam-619	216	8	the	the	DET
ejpam-619	216	9	feasible	feasible	ADJ
ejpam-619	216	10	solution	solution	NOUN
ejpam-619	216	11	x̄	x̄	PROPN
ejpam-619	216	12	of	of	ADP
ejpam-619	216	13	(	(	PUNCT
ejpam-619	216	14	fp	fp	NOUN
ejpam-619	216	15	)	)	PUNCT
ejpam-619	216	16	and	and	CCONJ
ejpam-619	216	17	all	all	DET
ejpam-619	216	18	feasible	feasible	ADJ
ejpam-619	216	19	solutions	solution	NOUN
ejpam-619	216	20	(	(	PUNCT
ejpam-619	216	21	u	u	NOUN
ejpam-619	216	22	,	,	PUNCT
ejpam-619	216	23	v	v	NOUN
ejpam-619	216	24	)	)	PUNCT
ejpam-619	216	25	of	of	ADP
ejpam-619	216	26	(	(	PUNCT
ejpam-619	216	27	wfd	wfd	NOUN
ejpam-619	216	28	)	)	PUNCT
ejpam-619	216	29	.	.	PUNCT
ejpam-619	217	1	then	then	ADV
ejpam-619	217	2	there	there	PRON
ejpam-619	217	3	exists	exist	VERB
ejpam-619	217	4	a	a	DET
ejpam-619	217	5	v̄	v̄	NOUN
ejpam-619	217	6	∈	∈	PROPN
ejpam-619	217	7	rm	rm	NOUN
ejpam-619	217	8	+	+	CCONJ
ejpam-619	217	9	such	such	ADJ
ejpam-619	217	10	that	that	SCONJ
ejpam-619	217	11	(	(	PUNCT
ejpam-619	217	12	x̄	x̄	NOUN
ejpam-619	217	13	,	,	PUNCT
ejpam-619	217	14	v̄	v̄	PROPN
ejpam-619	217	15	)	)	PUNCT
ejpam-619	217	16	is	be	AUX
ejpam-619	217	17	an	an	DET
ejpam-619	217	18	optimal	optimal	ADJ
ejpam-619	217	19	solution	solution	NOUN
ejpam-619	217	20	of	of	ADP
ejpam-619	217	21	(	(	PUNCT
ejpam-619	217	22	wfd	wfd	NOUN
ejpam-619	217	23	)	)	PUNCT
ejpam-619	217	24	and	and	CCONJ
ejpam-619	217	25	the	the	DET
ejpam-619	217	26	optimal	optimal	ADJ
ejpam-619	217	27	objective	objective	ADJ
ejpam-619	217	28	function	function	NOUN
ejpam-619	217	29	values	value	NOUN
ejpam-619	217	30	of	of	ADP
ejpam-619	217	31	(	(	PUNCT
ejpam-619	217	32	fp	fp	NOUN
ejpam-619	217	33	)	)	PUNCT
ejpam-619	217	34	and	and	CCONJ
ejpam-619	217	35	(	(	PUNCT
ejpam-619	217	36	wfd	wfd	NOUN
ejpam-619	217	37	)	)	PUNCT
ejpam-619	217	38	are	be	AUX
ejpam-619	217	39	equal	equal	ADJ
ejpam-619	217	40	.	.	PUNCT
ejpam-619	218	1	7	7	X
ejpam-619	218	2	.	.	X
ejpam-619	218	3	conclusion	conclusion	NOUN
ejpam-619	218	4	in	in	ADP
ejpam-619	218	5	this	this	DET
ejpam-619	218	6	paper	paper	NOUN
ejpam-619	218	7	a	a	DET
ejpam-619	218	8	new	new	ADJ
ejpam-619	218	9	concept	concept	NOUN
ejpam-619	218	10	of	of	ADP
ejpam-619	218	11	generalized	generalized	ADJ
ejpam-619	218	12	convexity	convexity	NOUN
ejpam-619	218	13	has	have	AUX
ejpam-619	218	14	been	be	AUX
ejpam-619	218	15	introduced	introduce	VERB
ejpam-619	218	16	.	.	PUNCT
ejpam-619	219	1	under	under	ADP
ejpam-619	219	2	this	this	DET
ejpam-619	219	3	generalized	generalized	ADJ
ejpam-619	219	4	convexity	convexity	NOUN
ejpam-619	219	5	we	we	PRON
ejpam-619	219	6	establish	establish	VERB
ejpam-619	219	7	sufficient	sufficient	ADJ
ejpam-619	219	8	optimality	optimality	NOUN
ejpam-619	219	9	conditions	condition	NOUN
ejpam-619	219	10	and	and	CCONJ
ejpam-619	219	11	duality	duality	NOUN
ejpam-619	219	12	results	result	NOUN
ejpam-619	219	13	for	for	ADP
ejpam-619	219	14	a	a	DET
ejpam-619	219	15	nonlinear	nonlinear	ADJ
ejpam-619	219	16	programming	programming	NOUN
ejpam-619	219	17	problem.these	problem.these	ADJ
ejpam-619	219	18	duality	duality	NOUN
ejpam-619	219	19	relations	relation	NOUN
ejpam-619	219	20	lead	lead	VERB
ejpam-619	219	21	to	to	ADP
ejpam-619	219	22	duality	duality	NOUN
ejpam-619	219	23	in	in	ADP
ejpam-619	219	24	nonlinear	nonlinear	ADJ
ejpam-619	219	25	fractional	fractional	ADJ
ejpam-619	219	26	programming	programming	NOUN
ejpam-619	219	27	.	.	PUNCT
ejpam-619	220	1	acknowledgements	acknowledgement	NOUN
ejpam-619	220	2	the	the	DET
ejpam-619	220	3	second	second	ADJ
ejpam-619	220	4	author	author	NOUN
ejpam-619	220	5	is	be	AUX
ejpam-619	220	6	thankful	thankful	ADJ
ejpam-619	220	7	to	to	ADP
ejpam-619	220	8	the	the	DET
ejpam-619	220	9	university	university	NOUN
ejpam-619	220	10	grants	grant	NOUN
ejpam-619	220	11	commission	commission	PROPN
ejpam-619	220	12	,	,	PUNCT
ejpam-619	220	13	new	new	ADJ
ejpam-619	220	14	delhi	delhi	PROPN
ejpam-619	220	15	(	(	PUNCT
ejpam-619	220	16	india	india	PROPN
ejpam-619	220	17	)	)	PUNCT
ejpam-619	220	18	for	for	ADP
ejpam-619	220	19	providing	provide	VERB
ejpam-619	220	20	financial	financial	ADJ
ejpam-619	220	21	support	support	NOUN
ejpam-619	220	22	during	during	ADP
ejpam-619	220	23	this	this	DET
ejpam-619	220	24	work	work	NOUN
ejpam-619	220	25	.	.	PUNCT
ejpam-619	221	1	references	reference	NOUN
ejpam-619	221	2	[	[	X
ejpam-619	221	3	1	1	NUM
ejpam-619	221	4	]	]	X
ejpam-619	221	5	i.	i.	PROPN
ejpam-619	221	6	ahmad	ahmad	PROPN
ejpam-619	221	7	and	and	CCONJ
ejpam-619	221	8	z.	z.	PROPN
ejpam-619	221	9	husain	husain	PROPN
ejpam-619	221	10	.	.	PUNCT
ejpam-619	222	1	second	second	ADJ
ejpam-619	222	2	-	-	PUNCT
ejpam-619	222	3	order	order	NOUN
ejpam-619	222	4	(	(	PUNCT
ejpam-619	222	5	f	f	X
ejpam-619	222	6	,	,	PUNCT
ejpam-619	222	7	α	α	PROPN
ejpam-619	222	8	,	,	PUNCT
ejpam-619	222	9	ρ	ρ	NOUN
ejpam-619	222	10	,	,	PUNCT
ejpam-619	222	11	d)-convexity	d)-convexity	NOUN
ejpam-619	222	12	and	and	CCONJ
ejpam-619	222	13	duality	duality	NOUN
ejpam-619	222	14	in	in	ADP
ejpam-619	222	15	multiobjective	multiobjective	ADJ
ejpam-619	222	16	programming	programming	NOUN
ejpam-619	222	17	.	.	PUNCT
ejpam-619	223	1	information	information	NOUN
ejpam-619	223	2	sciences	sciences	PROPN
ejpam-619	223	3	,	,	PUNCT
ejpam-619	223	4	176	176	NUM
ejpam-619	223	5	:	:	PUNCT
ejpam-619	223	6	3094	3094	NUM
ejpam-619	223	7	-	-	SYM
ejpam-619	223	8	3103	3103	NUM
ejpam-619	223	9	,	,	PUNCT
ejpam-619	223	10	2006	2006	NUM
ejpam-619	223	11	.	.	PUNCT
ejpam-619	224	1	[	[	X
ejpam-619	224	2	2	2	NUM
ejpam-619	224	3	]	]	PUNCT
ejpam-619	224	4	m.	m.	NOUN
ejpam-619	224	5	a.	a.	PROPN
ejpam-619	224	6	hanson	hanson	PROPN
ejpam-619	224	7	.	.	PUNCT
ejpam-619	225	1	on	on	ADP
ejpam-619	225	2	sufficiency	sufficiency	NOUN
ejpam-619	225	3	of	of	ADP
ejpam-619	225	4	the	the	DET
ejpam-619	225	5	kuhn	kuhn	PROPN
ejpam-619	225	6	-	-	PUNCT
ejpam-619	225	7	tucker	tucker	PROPN
ejpam-619	225	8	conditions	condition	NOUN
ejpam-619	225	9	.	.	PUNCT
ejpam-619	226	1	journal	journal	NOUN
ejpam-619	226	2	of	of	ADP
ejpam-619	226	3	mathematical	mathematical	ADJ
ejpam-619	226	4	analysis	analysis	NOUN
ejpam-619	226	5	and	and	CCONJ
ejpam-619	226	6	applications	application	NOUN
ejpam-619	226	7	,	,	PUNCT
ejpam-619	226	8	80	80	NUM
ejpam-619	226	9	:	:	SYM
ejpam-619	226	10	545	545	NUM
ejpam-619	226	11	-	-	SYM
ejpam-619	226	12	550	550	NUM
ejpam-619	226	13	,	,	PUNCT
ejpam-619	226	14	1981	1981	NUM
ejpam-619	226	15	.	.	PUNCT
ejpam-619	227	1	[	[	X
ejpam-619	227	2	3	3	X
ejpam-619	227	3	]	]	PUNCT
ejpam-619	227	4	m.	m.	NOUN
ejpam-619	227	5	a.	a.	PROPN
ejpam-619	227	6	hanson	hanson	PROPN
ejpam-619	227	7	and	and	CCONJ
ejpam-619	227	8	b.	b.	PROPN
ejpam-619	227	9	mond	mond	PROPN
ejpam-619	227	10	.	.	PUNCT
ejpam-619	228	1	further	further	ADJ
ejpam-619	228	2	generalizations	generalization	NOUN
ejpam-619	228	3	of	of	ADP
ejpam-619	228	4	convexity	convexity	NOUN
ejpam-619	228	5	in	in	ADP
ejpam-619	228	6	mathematical	mathematical	ADJ
ejpam-619	228	7	programming	programming	NOUN
ejpam-619	228	8	.	.	PUNCT
ejpam-619	229	1	journal	journal	PROPN
ejpam-619	229	2	of	of	ADP
ejpam-619	229	3	information	information	NOUN
ejpam-619	229	4	and	and	CCONJ
ejpam-619	229	5	optimization	optimization	NOUN
ejpam-619	229	6	sciences	science	NOUN
ejpam-619	229	7	,	,	PUNCT
ejpam-619	229	8	3	3	NUM
ejpam-619	229	9	:	:	SYM
ejpam-619	229	10	25	25	NUM
ejpam-619	229	11	-	-	SYM
ejpam-619	229	12	32	32	NUM
ejpam-619	229	13	,	,	PUNCT
ejpam-619	229	14	1986	1986	NUM
ejpam-619	229	15	.	.	PUNCT
ejpam-619	230	1	references	reference	NOUN
ejpam-619	230	2	275	275	NUM
ejpam-619	231	1	[	[	X
ejpam-619	231	2	4	4	NUM
ejpam-619	231	3	]	]	PUNCT
ejpam-619	231	4	z.	z.	PROPN
ejpam-619	231	5	a.	a.	PROPN
ejpam-619	231	6	liang	liang	PROPN
ejpam-619	231	7	,	,	PUNCT
ejpam-619	231	8	h.	h.	PROPN
ejpam-619	231	9	x.	x.	PROPN
ejpam-619	231	10	huang	huang	PROPN
ejpam-619	231	11	and	and	CCONJ
ejpam-619	231	12	p.	p.	PROPN
ejpam-619	231	13	m.	m.	NOUN
ejpam-619	231	14	pardalos	pardalo	NOUN
ejpam-619	231	15	.	.	PUNCT
ejpam-619	232	1	optimality	optimality	NOUN
ejpam-619	232	2	conditions	condition	NOUN
ejpam-619	232	3	and	and	CCONJ
ejpam-619	232	4	duality	duality	NOUN
ejpam-619	232	5	for	for	ADP
ejpam-619	232	6	a	a	DET
ejpam-619	232	7	class	class	NOUN
ejpam-619	232	8	of	of	ADP
ejpam-619	232	9	nonlinear	nonlinear	ADJ
ejpam-619	232	10	fractional	fractional	ADJ
ejpam-619	232	11	programming	programming	NOUN
ejpam-619	232	12	problems	problem	NOUN
ejpam-619	232	13	.	.	PUNCT
ejpam-619	233	1	journal	journal	NOUN
ejpam-619	233	2	of	of	ADP
ejpam-619	233	3	optimization	optimization	NOUN
ejpam-619	233	4	theory	theory	NOUN
ejpam-619	233	5	and	and	CCONJ
ejpam-619	233	6	applications	application	NOUN
ejpam-619	233	7	,	,	PUNCT
ejpam-619	233	8	110	110	NUM
ejpam-619	233	9	:	:	PUNCT
ejpam-619	233	10	611	611	NUM
ejpam-619	233	11	-	-	SYM
ejpam-619	233	12	619	619	NUM
ejpam-619	233	13	,	,	PUNCT
ejpam-619	233	14	2001	2001	NUM
ejpam-619	233	15	.	.	PUNCT
ejpam-619	234	1	[	[	X
ejpam-619	234	2	5	5	X
ejpam-619	234	3	]	]	X
ejpam-619	234	4	o.	o.	PROPN
ejpam-619	234	5	l.	l.	PROPN
ejpam-619	234	6	mangasarian	mangasarian	PROPN
ejpam-619	234	7	.	.	PUNCT
ejpam-619	235	1	nonlinear	nonlinear	PROPN
ejpam-619	235	2	programming	programming	PROPN
ejpam-619	235	3	.	.	PUNCT
ejpam-619	236	1	mcgraw	mcgraw	PROPN
ejpam-619	236	2	hill	hill	PROPN
ejpam-619	236	3	,	,	PUNCT
ejpam-619	236	4	new	new	PROPN
ejpam-619	236	5	york	york	PROPN
ejpam-619	236	6	,	,	PUNCT
ejpam-619	236	7	ny	ny	PROPN
ejpam-619	236	8	,	,	PUNCT
ejpam-619	236	9	1969	1969	NUM
ejpam-619	236	10	.	.	PUNCT
ejpam-619	237	1	[	[	X
ejpam-619	237	2	6	6	NUM
ejpam-619	237	3	]	]	PUNCT
ejpam-619	237	4	s.	s.	PROPN
ejpam-619	237	5	pandey	pandey	PROPN
ejpam-619	237	6	.	.	PUNCT
ejpam-619	238	1	duality	duality	NOUN
ejpam-619	238	2	for	for	ADP
ejpam-619	238	3	multiobjective	multiobjective	ADJ
ejpam-619	238	4	fractional	fractional	ADJ
ejpam-619	238	5	programming	programming	NOUN
ejpam-619	238	6	involving	involve	VERB
ejpam-619	238	7	generalized	generalized	ADJ
ejpam-619	238	8	ηbonvex	ηbonvex	NOUN
ejpam-619	238	9	functions	function	NOUN
ejpam-619	238	10	.	.	PUNCT
ejpam-619	239	1	opsearch	opsearch	PROPN
ejpam-619	239	2	,	,	PUNCT
ejpam-619	239	3	28	28	NUM
ejpam-619	239	4	:	:	SYM
ejpam-619	239	5	31	31	NUM
ejpam-619	239	6	-	-	SYM
ejpam-619	239	7	43	43	NUM
ejpam-619	239	8	,	,	PUNCT
ejpam-619	239	9	1991	1991	NUM
ejpam-619	239	10	.	.	PUNCT
ejpam-619	240	1	[	[	X
ejpam-619	240	2	7	7	X
ejpam-619	240	3	]	]	X
ejpam-619	240	4	v.	v.	X
ejpam-619	240	5	preda	preda	PROPN
ejpam-619	240	6	.	.	PUNCT
ejpam-619	241	1	on	on	ADP
ejpam-619	241	2	efficiency	efficiency	NOUN
ejpam-619	241	3	and	and	CCONJ
ejpam-619	241	4	duality	duality	NOUN
ejpam-619	241	5	for	for	ADP
ejpam-619	241	6	multiobjective	multiobjective	ADJ
ejpam-619	241	7	programs	program	NOUN
ejpam-619	241	8	.	.	PUNCT
ejpam-619	242	1	journal	journal	PROPN
ejpam-619	242	2	of	of	ADP
ejpam-619	242	3	mathematical	mathematical	ADJ
ejpam-619	242	4	analysis	analysis	NOUN
ejpam-619	242	5	and	and	CCONJ
ejpam-619	242	6	applications	application	NOUN
ejpam-619	242	7	,	,	PUNCT
ejpam-619	242	8	166	166	NUM
ejpam-619	242	9	:	:	SYM
ejpam-619	242	10	365	365	NUM
ejpam-619	242	11	-	-	SYM
ejpam-619	242	12	377	377	NUM
ejpam-619	242	13	,	,	PUNCT
ejpam-619	242	14	1992	1992	NUM
ejpam-619	242	15	.	.	PUNCT
ejpam-619	243	1	[	[	X
ejpam-619	243	2	8	8	X
ejpam-619	243	3	]	]	PUNCT
ejpam-619	243	4	j.	j.	PROPN
ejpam-619	243	5	p.	p.	PROPN
ejpam-619	243	6	vial	vial	PROPN
ejpam-619	243	7	.	.	PUNCT
ejpam-619	244	1	strong	strong	ADJ
ejpam-619	244	2	and	and	CCONJ
ejpam-619	244	3	weak	weak	ADJ
ejpam-619	244	4	convexity	convexity	NOUN
ejpam-619	244	5	of	of	ADP
ejpam-619	244	6	sets	set	NOUN
ejpam-619	244	7	and	and	CCONJ
ejpam-619	244	8	functions	function	NOUN
ejpam-619	244	9	.	.	PUNCT
ejpam-619	245	1	mathematics	mathematic	NOUN
ejpam-619	245	2	of	of	ADP
ejpam-619	245	3	operations	operation	NOUN
ejpam-619	245	4	research	research	NOUN
ejpam-619	245	5	,	,	PUNCT
ejpam-619	245	6	8	8	NUM
ejpam-619	245	7	:	:	SYM
ejpam-619	245	8	231	231	NUM
ejpam-619	245	9	-	-	SYM
ejpam-619	245	10	259	259	NUM
ejpam-619	245	11	,	,	PUNCT
ejpam-619	245	12	1983	1983	NUM
ejpam-619	245	13	.	.	PUNCT
ejpam-619	246	1	[	[	X
ejpam-619	246	2	9	9	NUM
ejpam-619	246	3	]	]	X
ejpam-619	246	4	d.	d.	PROPN
ejpam-619	246	5	h.	h.	PROPN
ejpam-619	246	6	yuan	yuan	PROPN
ejpam-619	246	7	,	,	PUNCT
ejpam-619	246	8	x.	x.	PROPN
ejpam-619	246	9	l.	l.	PROPN
ejpam-619	246	10	liu	liu	PROPN
ejpam-619	246	11	,	,	PUNCT
ejpam-619	246	12	a.	a.	NOUN
ejpam-619	246	13	chinchuluun	chinchuluun	PROPN
ejpam-619	246	14	and	and	CCONJ
ejpam-619	246	15	p.	p.	PROPN
ejpam-619	246	16	m.	m.	NOUN
ejpam-619	246	17	pardalos	pardalo	NOUN
ejpam-619	246	18	.	.	PUNCT
ejpam-619	247	1	nondifferentiable	nondifferentiable	ADJ
ejpam-619	247	2	minimax	minimax	NOUN
ejpam-619	247	3	fractional	fractional	ADJ
ejpam-619	247	4	programming	programming	NOUN
ejpam-619	247	5	problems	problem	NOUN
ejpam-619	247	6	with	with	ADP
ejpam-619	247	7	(	(	PUNCT
ejpam-619	247	8	c	c	NOUN
ejpam-619	247	9	,	,	PUNCT
ejpam-619	247	10	α	α	PROPN
ejpam-619	247	11	,	,	PUNCT
ejpam-619	247	12	ρ	ρ	NOUN
ejpam-619	247	13	,	,	PUNCT
ejpam-619	247	14	d)-convexity	d)-convexity	NOUN
ejpam-619	247	15	.	.	PUNCT
ejpam-619	248	1	journal	journal	PROPN
ejpam-619	248	2	of	of	ADP
ejpam-619	248	3	optimization	optimization	NOUN
ejpam-619	248	4	theory	theory	NOUN
ejpam-619	248	5	and	and	CCONJ
ejpam-619	248	6	applications	application	NOUN
ejpam-619	248	7	,	,	PUNCT
ejpam-619	248	8	129	129	NUM
ejpam-619	248	9	:	:	SYM
ejpam-619	248	10	185	185	NUM
ejpam-619	248	11	-	-	SYM
ejpam-619	248	12	199	199	NUM
ejpam-619	248	13	,	,	PUNCT
ejpam-619	248	14	2006	2006	NUM
ejpam-619	248	15	.	.	PUNCT