

CUBO A Mathematical Journal
Vol.14, No¯ 02, (01–13). June 2012

The ǫ−Optimality Conditions for Multiple Objective
Fractional Programming Problems for Generalized

(ρ, η)−Invexity of Higher Order

Ram U. Verma

International Publications,

3400 S Brahma Blvd, Suite 31B,

Kingsville, Texas 78363, USA

email: verma99@msn.com

ABSTRACT

Motivated by the recent investigations in literature, a general framework for a class

of (ρ, η)−invex n-set functions of higher order is introduced, and then some results on

the ǫ−optimality conditions for multiple objective fractional subset programming are

explored. The obtained results are general in nature, while generalize and unify results

on generalized invexity as well as on generalized invexity of higher order to the context

of multiple fractional programming.

RESUMEN

Motivado por investigaciones recientes en la literatura, se introduce un marco gen-

eral para una clase de funciones (ρ, η)−invex n-set de orden superior y se exploran

algunos resultados sobre condiciones de épsilon-optimalidad para objetivos múltiples

fraccionales de subconjuntos de programación. Los resultados obtenidos son de natu-

raleza general, dado que generalizan y unifican resultados sobre invexity generalizada

e invexity generalizada de orden superior en el contexto de la programación múltiple

fraccionaria.

Keywords and Phrases: Generalized invexity of higher order, multiple objective fractional

subset programming, ǫ−efficient solution, semi-parametric sufficient ǫ−optimality conditions.

2010 AMS Mathematics Subject Classification: 49J40, 90C25.


2 Ram U. Verma CUBO
14, 2 (2012)

1 Introduction

Recently, Kim et al. [9] investigated some results based on ǫ−optimality conditions for multiple

objective fractional optimization problems. They used both approaches of parametric as well as

non-parametric sufficient conditions to achieving an equivalence between them. Motivated by these

developments, we examine some ǫ−optimality conditions for multiple objective fractional program-

ming problems based on a generalized (ρ, η)−invexity for higher order [1,7,8] of n-set functions,

more specifically, results on parametric and semi-parametric sufficient ǫ−efficiency conditions for

multiobjective fractional subset programming. More recently, Mishra et al. [11] published some

results on optimality conditions for multiple objective fractional subset programming with invex

and related non-convex n-set functions (also studied by Verma [13,15] and Zalmai [16]) to the case

of parametric and semi-parametric sufficient efficiency conditions for a multiobjective fractional

subset programming problem. Jeyakumar et al. [5,6] and Kim et al. [9] investigated some results

on ǫ−optimality conditions for multiobjective fractional programming problems. We present using

the generalized invexity of higher order for differentiable functions, the following multiple objective

fractional subset programming problem:

(P)

Minimize (
F1(S)

G1(S)
,
F2(S)

G2(S)
, ...,

Fp(S)

Gp(S)
)

subject to

Hj(S) ≤ 0 for j ∈ {1, ..., m}, S ∈ Q = {x ∈ X : Hj(x) ≤ 0, j ∈ {1, ..., m}},

where X is an open convex subset of ℜn (n-dimensional Euclidean space), Fi, Gi, i ∈ {1, ..., p} and

Hj for j ∈ {1, ..., m} are real-valued functions defined on X and Gi(S) > 0 for each i ∈ {1, ..., p} and

for all S ∈ X.

Next, we observe that problem (P) is equivalent to the parametric multiobjective non-fractional

programming problem:

(Pλ)

Minimize (F1(S) − λ1G1(S), ..., Fp(S) − λpGp(S)),

where λi, i = 1, 2, ..., p are parameters, and S ∈ Q.

Mishra et al. [11] investigated several parametric and semi-parametric sufficient conditions

for the multiobjective fractional subset programming problems based on generalized invexity as-

sumptions. Moreover, these results are also applicable to other classes of problems with multiple,

fractional, and conventional objective functions.

Furthermore, among other results, the obtained results generalize the recent results on generalized


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The ǫ−Optimality Conditions ... 3

invexity to the case of the generalized invexity of higher order m ≥ 1 relating to the case of semi-

parametric sufficient ǫ−efficiency conditions for the multiobjective fractional subset programming

problems. For more details, we refer the reader [1–17].

2 Generalized Invexities of Higher Order

In this section, we develop some concepts and notations for the problem on hand. Let X be an

open convex subset of ℜn (n-dimensional Euclidean space). Let 〈·, ·〉 the inner product, and let

η : X × X → ℜn be a vector-valued function. Suppose that ▽f(x)denotes the gradient of f at x

defined by

▽f(x) = (
∂f(x)

∂x1
, ...,

∂f(x)

∂xn
),

where f : X → ℜn is real-valued function on X.

Next, we recall the notions of the generalized invexity. Let S, S∗ ∈ X, let the function

F : X → ℜn with components Fi for i ∈ {1, ..., n}, be differentiable at S
∗.

Definition 2.1. A differentiable function F : X → ℜn is said to be (ρ, η)−invex of higher order at

S∗ if there exists a vector-valued function η : X × X → ℜn such that for each S∗ ∈ X, and ρ > 0,

F(S) − F(S∗) ≥ 〈▽F(S∗), η(S, S∗)〉 + ρ‖S − S∗‖m,

where m ≥ 1 is an integer.

Definition 2.2. A differentiable function F : X → ℜn is said to be (ρ, η)−strictly-invex of higher

order at S∗ if there exists a vector-valued function η : X × X → ℜn such that for each S∗ ∈ X, and

ρ > 0,

F(S) − F(S∗) > 〈▽F(S∗), η(S, S∗)〉 + ρ‖S − S∗‖m,

where m ≥ 1 is an integer.

Definition 2.3. A differentiable function F : X → ℜn is said to be (ρ, η)−quasi-invex of higher

order at S∗ if there exists a vector-valued function η : X × X → ℜn such that for each S∗ ∈ X, and

ρ > 0,

F(S) ≤ F(S∗) ⇒ 〈▽F(S∗), η(S, S∗)〉 + ρ‖S − S∗‖m ≤ 0,

where m ≥ 1 is an integer.

Definition 2.4. A differentiable function F : X → ℜn is said to be (ρ, η)−prestrictly-quasi-invex

of higher order at S∗ if there exists a vector-valued function η : X × X → ℜn such that for each


4 Ram U. Verma CUBO
14, 2 (2012)

S∗ ∈ X, and ρ > 0,

F(S) < F(S∗) ⇒ 〈▽F(S∗), η(S, S∗)〉 + ρ‖S − S∗‖m ≤ 0,

where m ≥ 1 is an integer.

Definition 2.5. A differentiable function F : X → ℜn is said to be (ρ, η)−pseudo-invex of higher

order at S∗ if there exists a vector-valued function η : X × X → ℜn such that for each S∗ ∈ X, and

ρ > 0,

〈▽F(S∗), η(S, S∗)〉 + ρ‖S − S∗‖m ≥ 0 ⇒ F(S) ≥ F(S∗),

where m ≥ 1 is an integer.

Definition 2.6. A differentiable function F : X → ℜn is said to be (ρ, η)−strictly-pseudo-invex

of higher order at S∗ if there exists a vector-valued function η : X × X → ℜn such that for each

S∗ ∈ X, and ρ > 0,

〈▽F(S∗), η(S, S∗)〉 + ρ‖S − S∗‖m ≥ 0 ⇒ F(S) > F(S∗),

where m ≥ 1 is an integer.

Definition 2.7. A differentiable function F : X → ℜn is said to be (ρ, η)−prestrictly-pseudo-invex

of higher order at S∗ if there exists a vector-valued function η : X × X → ℜn such that for each

S∗ ∈ X, and ρ > 0,

〈▽F(S∗), η(S, S∗)〉 + ρ‖S − S∗‖m > 0 ⇒ F(S) ≥ F(S∗),

where m ≥ 1 is an integer.

Definition 2.8. A differentiable function F : X → ℜn is said to be (ρ, η)−strictly-quasi-invex

of higher order at S∗ if there exists a vector-valued function η : X × X → ℜn such that for each

S∗ ∈ X, and ρ > 0,

F(S) ≤ F(S∗) ⇒ 〈▽F(S∗), η(S, S∗)〉 + ρ‖S − S∗‖m < 0,

where m ≥ 1 is an integer.

Definition 2.9. A differentiable function F : X → ℜn is said to be (ρ, η)−prestrictly-quasi-invex

of higher order at S∗ if there exists a vector-valued function η : X × X → ℜn such that for each

S∗ ∈ X, and ρ > 0,

F(S) < F(S∗) ⇒ 〈▽F(S∗), η(S, S∗)〉 + ρ‖S − S∗‖m ≤ 0,


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The ǫ−Optimality Conditions ... 5

where m ≥ 1 is an integer.

Now we introduce the generalized ǫ−solvability conditions for (P) and (Pλ) problems as

follows: S∗ is a generalized ǫ−efficient solution to (P) if there does not exist an S ∈ Q such that

Fi(S)

Gi(S)
≤

Fi(S
∗)

Gi(S
∗)

− ǫi(S
∗) ∀ i = 1, .., p,

Fj(S)

Gj(S)
<

Fj(S
∗)

Gj(S
∗)

− ǫj(S
∗) for some j ∈ {1, .., p},

where ǫi,ǫj : ℜ
n
→ ℜ are with ǫ(S∗) = (ǫ1(S

∗), ..., ǫp(S
∗)), ǫi ≥ 0 for i=1,...,p.

For ǫ = ǫ(S∗), (P) reduces to Kim et al. [9], and for ǫ = 0, it reduces to the case that S∗ ∈ Q is

an efficient solution to (P) if there exists no S ∈ Q such that

Fi(S)

Gi(S)
≤

Fi(S
∗)

Gi(S
∗)

∀ i = 1, ..., p.

To this context, based on Mishra et al. [11], we consider the following auxiliary problem:

(Pλ)

minimizeS∈Q(F1(S) − λ1G1(S), ..., Fp(S) − λpGp(S)),

where λi for i ∈ {1, ..., p} are parameters.

Next, we introduce the generalized ǫ−solvability conditions for (Pλ) problem as follows: S∗ is

a generalized ǫ−efficient solution to (Pλ) if there does not exist an S ∈ Q such that

Fi(S) − λiGi(S) ≤ Fi(S
∗) − λiGi(S

∗) − ǭi(S
∗) ∀ i = 1, .., p,

Fj(S) − λjGj(S) < Fj(S
∗
) − λjGj(S

∗
) − ǭj(S

∗
) for some j ∈ {1, .., p},

where λi =
Fi(S

∗

)

Gi(S
∗)

− ǫi, ǭi(S
∗) = ǫi(S

∗)Gi(S
∗), and ǭj(S

∗) = ǫj(S
∗)Gj(S

∗), where ǫi,ǫj : ℜ
n
→ ℜ

are with ǫ(S∗) = (ǫ1(S
∗), ..., ǫp(S

∗)), ǫi ≥ 0 for i=1,...,p.

For ǫ = ǫ(S∗), (P) reduces to Kim et al. [9], and for ǫ = 0, it reduces to the case that is an efficient

solution to (P) if there exists no S ∈ Ξ such that

(
F1(S)

G1(S)
,
F2(S)

G2(S)
, ...,

Fp(S)

Gp(S)
) ≤ (

F1(S
∗)

G1(S
∗)
,
F2(S

∗)

G2(S
∗)
, ...,

Fp(S
∗)

Gp(S
∗)
).

Lemma 2.1. [9] Let S∗ ∈ Q = {x ∈ X : Hj(x) ≤ 0 for i = 1, ..., m}, where Hj : X → ℜ is a

real-valued function on X. Then the following statements are mutually equivalent:

(i) S∗ is a generalized ǫ(S∗)−efficient solution to (P).


6 Ram U. Verma CUBO
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(ii) S∗ is a generalized ǫ∗(S∗)−solution to (Pλ), where

λ = (
F1(S

∗)

G1(S
∗)

− ǫ1(S
∗
), ...,

Fp(S
∗)

Gp(S
∗)

− ǫp(S
∗
))

and ǫ∗(S∗) = (ǫ1(S
∗)G1(S

∗), ..., ǫp(S
∗)Gp(S

∗)).

Lemma 2.2. [15] Let S∗ ∈ Q={x ∈ X : Hj(x) ≤ 0 for i = 1, ..., m}, where Hj : X → ℜ is a

real-valued function on X. Then the following statements are mutually equivalent:

(i) S∗ is a generalized ǫ(S∗)−efficient solution to (P).

(ii) There exists S ∈ Q such that

Σ
p
i=1

[Fi(S) − (
Fi(S

∗)

Gi(S
∗)

− ǫi(S
∗
))Gi(S)] ≥ 0.

Lemma 2.3. [15] Let S∗ ∈ Q = {x ∈ X : Hj(x) ≤ 0 for i = 1, ..., m}, where Hj : X → ℜ is a

real-valued function on X. Then the following statements are mutually equivalent:

(i) S∗ is a generalized ǫ(S∗)−efficient solution to (Pλ).

(ii) There exists S ∈ Q such that

Σ
p
i=1

[Fi(S)−(
Fi(S

∗)

Gi(S)
−ǫi(S

∗))Gi(S)] ≥ Σ
p
i=1

[Fi(S
∗)−(

Fi(S
∗)

Gi(S
∗)
−ǫi(S

∗))Gi(S
∗)]−Σ

p
i=1

ǫi(S
∗)Gi(S

∗).

3 Parametric Sufficient ǫ− Optimality Conditions

This section deals with some parametric sufficient ǫ− optimality conditions for problem (P)

under the generalized frameworks for generalized (ρ, η)−invexity of higher order m ≥ 1. We begin

with real-valued functions Ai(.; λ, u) and Bj(., v) defined by

Ai(.; λ, u) = ui[Fi(S) − λiGi(S)] for i = 1, ..., p, and for fixed λ, u and v

and

Bj(., v) = vjHj(S), j = 1, ..., m.

Theorem 3.1. Let S∗ ∈ Q = {S ∈ X : Hj(S) ≤ 0 for j ∈ {1, ..., m}, the feasible set of (P).

Let Fi, Gi, i ∈ {1, ..., p}, and Hj, j ∈ {1, ..., m}, be differentiable at S
∗ ∈ Q, and let there exist

u∗ ∈ U = {u ∈ ℜp : u > 0, Σ
p
i=1

ui = 1} and v
∗ ∈ Rm+ such that

〈Σ
p
i=1

u∗i [▽Fi(S
∗
) − λ∗i ▽ Gi(S

∗
)] + Σmj=1v

∗

j ▽ Hj(S
∗
), η(S, S∗)〉 ≥ 0 ∀S ∈ Q, (3.1)


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The ǫ−Optimality Conditions ... 7

Fi(S
∗) − λ∗i Gi(S

∗) = 0 for i ∈ {1, ..., p}, (3.2)

v∗j Hj(S
∗
) = 0 for j ∈ {1, ..., m}, (3.3)

where λ∗i = (
Fi(S

∗

)

Gi(S
∗)

− ǫi(S
∗)).

Suppose, in addition, that any one of the following assumptions holds:

(i) Ai(.; λ
∗, u∗) (∀i = 1, ..., p) are(ρ, η)−pseudo-invex at S∗ of higher order and Bj(., v

∗) ∀j ∈

{1, ..., m} are (ρ, η)−quasi-invex at S∗ of higher order.

(ii) Ai(.; λ
∗, u∗) (∀i ∈ {1, ..., p} are (ρ, η)−prestrictly-pseudo-invex at S∗ of higher order and

Bj(., v
∗) ∀j ∈ {1, ..., m} are strictly-quasi-invex at S∗ of higher order.

(iii) Ai(.; λ
∗, u∗) (∀i ∈ {1, ..., p} are (ρ, η)−prestrictly-quasi-invex at S∗ of higher order and

Bj(., v
∗) ∀j ∈ {1, ..., m} are (ρ, η)−strictly-pseudo-invex at S∗ of higher order.

Then S∗ is an ǫ−efficient solution to (P).

Proof. If (i) holds, and if S∗ ∈ Q, then it follows from (3.1) that

〈Σ
p
i=1

u∗i [▽Fi(S
∗
) − λ∗i ▽ Gi(S

∗
)], η(S, S∗〉

+ 〈Σmj=1v
∗

j ▽ Hj(S
∗), η(S, S∗)〉 ≥ 0 ∀S ∈ Q. (3.4)

Since v∗ ≥ 0, S ∈ Q and (3.3) holds, we have

Σmj=1v
∗

j Hj(S) ≤ 0 = Σ
m
j=1v

∗

j Hj(S
∗),

and in light of the (ρ, η)−quasi-invexity of Bj(., v
∗) at S∗, we arrive at

〈Σmj=1v
∗

j ▽ Hj(S
∗), η(S, S∗)〉 ≤ −ρ‖S − S∗‖m. (3.5)

It follows from (3.4) and (3.5) that

〈Σ
p
i=1

u∗i [▽Fi(S
∗
) − λ∗i ▽ Gi(S

∗
)], η(S, S∗〉 ≥ ρ‖S − S∗‖m. (3.6)

This further implies

〈Σ
p
i=1

u∗i [▽Fi(S
∗
) − λ∗i ▽ Gi(S

∗
)], η(S, S∗〉 ≥ −ρ‖S − S∗‖m. (3.7)

Next, applying the (ρ, η)−pseudo-invexity at S∗ to (3.6), we have

Σ
p
i=1

u∗i [Fi(S) − λ
∗

i Gi(S)] ≥ Σ
p
i=1

u∗i [Fi(S
∗
) − λ∗i Gi(S

∗
)],


8 Ram U. Verma CUBO
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that is equivalent to

Σ
p
i=1

u∗i [Fi(S) − λ
∗

i Gi(S)] ≥ Σ
p
i=1

u∗i [Fi(S
∗) − λ∗i Gi(S

∗)]

− Σ
p
i=1

u∗i ǫi(S
∗)Gi(S

∗)

= 0,

where λ∗i = (
Fi(S

∗

)

Gi(S
∗)

− ǫi(S
∗)). Thus, we have

Σ
p
i=1

u∗i [Fi(S) − λ
∗

i Gi(S)] ≥ 0. (3.8)

Since u∗i > 0 for each i ∈ {1, ..., p}, we conclude that there does not exist an S ∈ Q such that

Fi(S)

Gi(S)
− (

Fi(S
∗)

Gi(S
∗)

− ǫi(S
∗)) ≤ 0 ∀ i = 1, ..., p,

Fj(S)

Gj(S)
− (

Fj(S
∗)

Gj(S
∗)

− ǫj(S
∗
)) < 0 ∀ j ∈ {1, ..., p}.

Hence, S∗ is an ǫ−efficient solution to (P). Similar proofs hold for (ii) and (iii).

When m=2, we have

Theorem 3.2. Let S∗ ∈ Q = {S ∈ X : Hj(S) ≤ 0 for j ∈ {1, ..., m}, the feasible set of (P).

Let Fi, Gi, i ∈ {1, ..., p}, and Hj, j ∈ {1, ..., m}, be differentiable at S
∗ ∈ Q, and let there exist

u∗ ∈ U = {u ∈ ℜp : u > 0, Σ
p
i=1ui = 1} and v

∗ ∈ Rm+ such that

〈Σ
p
i=1

u∗i [▽Fi(S
∗
) − λ∗i ▽ Gi(S

∗
)] + Σmj=1v

∗

j ▽ Hj(S
∗
), η(S, S∗)〉 ≥ 0 ∀S ∈ Q, (3.9)

Fi(S
∗) − λ∗i Gi(S

∗) = 0 for i ∈ {1, ..., p}, (3.10)

v∗j Hj(S
∗
) = 0 for j ∈ {1, ..., m}, (3.11)

where λ∗i = (
Fi(S

∗

)

Gi(S
∗)

− ǫi(S
∗)).

Suppose, in addition, that any one of the following assumptions holds:

(i) Ai(.; λ
∗, u∗) (∀i = 1, ..., p) are (ρ, η)−pseudo-invex at S∗ and Bj(., v

∗) ∀j ∈ {1, ..., m} are

(ρ, η)−quasi-invex at S∗.

(ii) Ai(.; λ
∗, u∗) (∀i ∈ {1, ..., p} are (ρ, η)−prestrictly-pseudo-invex at S∗ and Bj(., v

∗) ∀j ∈

{1, ..., m} are strictly-quasi-invex at S∗.


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The ǫ−Optimality Conditions ... 9

(iii) Ai(.; λ
∗, u∗) (∀i ∈ {1, ..., p} are (ρ, η)−prestrictly-quasi-invex at S∗ and Bj(., v

∗) ∀j ∈ {1, ..., m}

are (ρ, η)−strictly-pseudo-invex at S∗.

Then S∗ is an ǫ−efficient solution to (P).

For ǫ = 0, we have

Theorem 3.3. Let S∗ ∈ Q, let Fi, Gi, i ∈ {1, ..., p}, and Hj, j ∈ {1, ..., m}, be differentiable at

S∗ ∈ Λ, and let there exist u∗ ∈ U = {u ∈ ℜp : u > 0, Σ
p
i=1

ui = 1} and v
∗ ∈ Rm+ such that

〈Σ
p
i=1u

∗

i [▽Fi(S
∗
) − λ∗i ▽ Gi(S

∗
)] + Σmj=1v

∗

j ▽ Hj(S
∗
), η(S, S∗)〉

+ ρ‖S − S∗‖2 ≥ 0 ∀S ∈ Λn, (3.12)

Fi(S
∗) − λ∗i Gi(S

∗) = 0 for i ∈ {1, ..., p}, (3.13)

v∗j Hj(S
∗) = 0 for j ∈ {1, ..., m}. (3.14)

Suppose, in addition, that any one of the following assumptions holds:

(i) Ai(.; λ
∗, u∗) (∀i = 1, ..., p) are (ρ, η)−pseudo-invex at S∗ of higher order and Bj(., v

∗) (∀j ∈

{1, ..., m} are (ρ, η)−quasi-invex at S∗ of order.

(ii) Ai(.; λ
∗, u∗) (∀i ∈ {1, ..., p} are (ρ, η)−prestrictly-pseudo-invex at S∗ of higher order and

Bj(., v
∗) (∀j ∈ {1, ..., m} are(ρ, η)−strictly-quasi-invex at S∗ of higher order.

(iii) Ai(.; λ
∗, u∗) (∀i ∈ {1, ..., p} are (ρ, η)−prestrictly-quasi-invex at S∗ of higher order and Bj(., v

∗)

(∀j ∈ {1, ..., m} are (ρ, η)−strictly-pseudo-invex at S∗ of higher order.

Then S∗ is an efficient solution to (P).

Theorem 3.4. ([11], Theorem 3.1) Let S∗ ∈ Q, let Fi, Gi, i ∈ {1, ..., p}, and Hj, j ∈ {1, ..., m}, be

differentiable at S∗ ∈ Q, and let there exist u∗ ∈ U = {u ∈ ℜp : u > 0, Σ
p
i=1ui = 1} and v

∗ ∈ Rm+
such that

〈Σ
p
i=1u

∗

i [▽Fi(S
∗
) − λ∗i ▽ Gi(S

∗
)] + Σmj=1v

∗

j ▽ Hj(S
∗
), η(S, S∗)〉

+ ρ‖S − S∗‖2 ≥ 0 ∀S ∈ Λn, (3.15)


10 Ram U. Verma CUBO
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Fi(S
∗) − λ∗i Gi(S

∗) = 0 for i ∈ {1, ..., p}, (3.16)

v∗j Hj(S
∗
) = 0 for j ∈ {1, ..., m}. (3.17)

Suppose, in addition, that any one of the following assumptions holds:

(i) Ai(.; λ
∗, u∗) (∀i = 1, ..., p) arepseudo-invex at S∗ and Bj(., v∗) (∀j ∈ {1, ..., m} are quasi-

invex at S∗.

(ii) Ai(.; λ
∗, u∗) (∀i ∈ {1, ..., p} are prestrictly-pseudo-invex at S∗ and Bj(., v

∗) (∀j ∈ {1, ..., m}

are strictly-quasi-invex at S∗.

(iii) Ai(.; λ
∗, u∗) (∀i ∈ {1, ..., p} are prestrictly-quasi-invex at S∗ and Bj(., v

∗) (∀j ∈ {1, ..., m} are

strictly-pseudo-invex at S∗.

Then S∗ is an efficient solution to (P).

4 Semi-Parametric Sufficient ǫ− Optimality Conditions

This section deals with some semi-parametric sufficient ǫ− optimality conditions for problem

(P) under the generalized frameworks for generalized invexity. We start with real-valued functions

Ei(., S
∗, u∗), Bj(., v), and Hi(., S

∗, u∗, v∗) defined by

Ei(S, S
∗, u∗) = ui[Fi(S) − (

Fi(S
∗)

Gi(S
∗)

− ǫi(S
∗))Gi(S)] for i ∈ {1, ..., p},

Li(S, S
∗, u∗, v∗) = u∗i [Fi(S) − (

Fi(S
∗)

Gi(S
∗)

− ǫi(S
∗))Gi(S)] + Σj∈J0v

∗

j Hj(S) for i ∈ {1, ..., p},

and

Bj(., v) = vjHj(S), j = 1, ..., m.

Theorem 4.1. Let S∗ ∈ Q = {S ∈ X : Hj(S) ≤ 0 for j ∈ {1, ..., m}, the feasible set of (P).

Let Fi, Gi, i ∈ {1, ..., p}, and Hj, j ∈ {1, ..., m}, be differentiable at S
∗ ∈ Q, and let there exist


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The ǫ−Optimality Conditions ... 11

u∗ ∈ U = {u ∈ ℜp : u > 0, Σ
p
i=1

ui = 1} and v
∗ ∈ Rm+ such that

〈Σ
p
i=1

u∗i [▽Fi(S
∗) − (

Fi(S
∗)

Gi(S
∗)

− ǫi(S
∗)) ▽ Gi(S

∗)] + Σj∈J0v
∗

j ▽ Hj(S
∗), η(S, S∗)〉 ≥ 0 , (4.1)

and

v∗j Hj(S
∗) = 0 for j ∈ {1, ..., m}. (4.2)

Suppose, in addition, that any one of the following assumptions holds:

(i) Ei(.; S
∗, u∗) (∀i = 1, ..., p) are (ρ, η)−pseudo-invex at S∗ of higher order and Bj(., v

∗) (∀j ∈

{1, ..., m} are (ρ, η)−quasi-invex at S∗ of higher order.

(ii) Ei(.; S
∗, u∗) (∀i ∈ {1, ..., p} are (ρ, η)−prestrictly-pseudo-invex at S∗ of higher order and

Bj(., v
∗) (∀j ∈ {1, ..., m} are(ρ, η)−strictly-quasi-invex at S∗ of higher order.

(iii) Ei(.; S
∗, u∗) (∀i ∈ {1, ..., p} are(ρ, η)−prestrictly-quasi-invex at S∗ of higher order and Bj(., v

∗)

(∀j ∈ {1, ..., m} are (ρ, η)−strictly-pseudo-invex at S∗ of higher order.

Then S∗ is an ǫ−efficient solution to (P).

Proof. If (i) holds, and if S ∈ Q, then it follows from (4.1) that

〈Σ
p
i=1

u∗i [▽Fi(S
∗
) − (

Fi(S
∗)

Gi(S
∗)

− ǫi(S
∗
)) ▽ Gi(S

∗
)], η(S, S∗〉

+ 〈Σmj=1v
∗

j ▽ Hj(S
∗), η(S, S∗)〉 ≥ 0 ∀S ∈ Λn. (4.3)

Since v∗ ≥ 0, S ∈ Q and (4.2) holds, we have

Σmj=1v
∗

j Hj(S) ≤ 0 = Σ
m
j=1v

∗

j Hj(S
∗),

and in light of the (ρ, η)−quasi-invexity of Bj(., v
∗) at S∗, we arrive at

〈Σmj=1v
∗

j ▽ Hj(S
∗), η(S, S∗)〉 ≤ −ρ‖S − S∗‖m. (4.4)

It follows from (4.3) and (4.4) that

〈Σ
p
i=1

u∗i [▽Fi(S
∗) − (

Fi(S
∗)

Gi(S
∗)

− ǫi(S
∗)) ▽ Gi(S

∗)], η(S, S∗〉 ≥ ρ‖S − S∗‖m. (4.5)

Next, applying the (ρ, η)−pseudo-invexity at S∗ to (4.5), we have

Σ
p
i=1

u∗i [Fi(S) − (
Fi(S

∗)

Gi(S
∗)

− ǫi(S
∗))Gi(S)] ≥ Σ

p
i=1

u∗i [Fi(S
∗) − (

Fi(S
∗)

Gi(S
∗)

− ǫi(S
∗))Gi(S

∗)],


12 Ram U. Verma CUBO
14, 2 (2012)

that is equivalent to

Σ
p
i=1

u∗i [Fi(S) − (
Fi(S

∗)

Gi(S
∗)

− ǫi(S
∗))Gi(S)] ≥ Σ

p
i=1

u∗i [Fi(S
∗) − (

Fi(S
∗)

Gi(S
∗)

− ǫi(S
∗))Gi(S

∗)]

− Σ
p
i=1

u∗i ǫiGi(S
∗)

= 0.

Thus, we have

Σ
p
i=1

u∗i [Fi(S) − (
Fi(S

∗)

Gi(S
∗)

− ǫi(S
∗))Gi(S)] ≥ 0. (4.6)

Since u∗i > 0 for each i ∈ {1, ..., p}, we conclude that there does not exist an S ∈ Q such that

F1(S)

G1(S)
− (

Fi(S
∗)

Gi(S
∗)

− ǫi(S
∗)) ≤ 0 ∀ i = 1, ..., p,

Fj(S)

Gj(S)
− (

Fj(S
∗)

Gj(S
∗)

− ǫj(S
∗)) < 0 ∀ j ∈ {1, ..., p}.

Hence, S∗ is an ǫ−efficient solution to (P). Similar proofs hold for (ii) and (iii).

Received: August 2011. Revised: August 2011.

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	Introduction
	Generalized Invexities of Higher Order
	Parametric Sufficient - Optimality Conditions
	Semi-Parametric Sufficient - Optimality Conditions