Roughly B invex Multi Objective Programming Problems

(1)

Roughly

B-

invex Multi-Objective Programming Problems

Tarek Emam

Department of Mathematics, Faculty of Science, Suez Canal University, Suez, Egypt Email: [email protected]

Received July 4, 2012; revised August 12, 2012; accepted August 31, 2012

ABSTRACT

In this paper, we shall be interested in characterization of efficient solutions for special classes of problems. These classes consider roughlyB-invexity of involved functions. Sufficient and necessary conditions for a feasible solution to be an efficient or properly efficient solution are obtained.

Keywords: Multi-Objective ProgrammingProblems; RoughlyB-invex; Efficient Solutions; Properly Efficient

Solutions

1. Introduction

The study of multi-objective programming problems was very active in recent years. The minimum (efficient, Pareto) solution is an important concept in mathematical models, economics, decision theory, optimal control and game theory (see, for example, [1]). In most works, an assumption of convexity was made for the objective functions. Very recently, some generalized convexity has received more attention (see, for example, [2-6]). A sig-nificant generalization of convex functions is invex func-tion introduced first by Hanson [7], which has greatly been applied in nonlinear optimization and other bran- ches of pure and applied sciences.

The concept of B-invex functions was proposed by [8] as generalization of convex functions; these functions were extended to quasi B-invex, and pseudo B-invex functions. Many functions seem to be B-invex, but they are not, and many non B-invex functions are able to get

B-invex by choosing a suitable condition. Based on the previous discussion, Tarek [9] introduced a new class of

B-invex functions, this class called roughly B-invex functions.

Inspired and motivated by above works, the purpose of this paper is to formulate a multi-objective programming problem which it involves roughly B-invex functions. An efficient solution for considered problem is characterized by weighting and ε-constraint approaches. In the end of

the paper, we obtain sufficient and necessary conditions for a feasible solution to be an efficient or properly effi-cient solution for this kind of problems. Let us survey, briefly, the definitions and some results of roughly B-invexity.

Definition 1 [10]

Let _{y M}_ __Rn_{. The set}_M_{is said to be}_B-_invex_with

respect to _:_{M M}_ __Rn_at_{y M}_ _{if there exists}



, ,



:

 

b x y  MM 0,1 R,

such that y b

 

x y, M , for each x M , and

0  1.

M is said to be B-invex set with respect to η if M is

B-invexat each y M with respect to the same η. Note that, as in convex set, the intersection of finite (or infinite) family of B-invex sets is B-invex but the union is not necessarily B-invex set. Also, the sum of B-invex sets and the multiplying a B-invex set by a real number are again B-invex sets. Every B-invexset with respect to

:_{M M}_ __Rn

n

 is an invex set when b = 1; but the converse is not necessarily true.

A numerical function f , defined on a B-invexsubset M

of Rn_{, is said to be roughly} _B-_invex _{with respect to}

:M M R

 with roughness degree r at y M if there exists b x y



, ,



:MM

 

0,1 R, such that

 



,



  

1

  

f y b x y bf x  b f y ,

for each x M , and 0  1 such that x y r.

f is said to be roughly B-invex on M with respect to

 

x y,

 if it is roughlyB-invex at each y M with respect to the same 

 

x y,

n

.

Every invex function, with respect to η is roughly

B-invex function with respect to same η, where b (x, y) = 1; but the converse is not necessarily true. If the func-tions f Ri : R

n

are all roughly B-invex with respect to :M M R with roughness degree rii on a

B-invex set _M__Rn

 

k

i

h x 





, then the function

 

i if x

1

a

(2)

degree

1mini k i

r r

 



on M for ai≥ 0. If f R: n 

n R is roughlyB-invexwith respect to :M M R

n

 with roughness degree r, on

B-invex set MR , then for any real number R

the level set K_ 



x x M:  ,f x

 





is B-invexset. A numerical function f defined on a B-invex set

n

MR

:

is roughly B-invex function with respect to n

M M R

 with roughness degree r if and only if

epi(f) is a B-invex set. If

 

fi _{i I}_ is a family of numerical

functions, which are roughly B-invex with respect to

:_{M M}_ __Rn

 with roughness degree r and bounded from above on a B-invex set _M __Rn_{, then the}

numeri-cal function

 

i

 

i I

f x Sup f x





is a roughly B-invex with respect to same η with rough-ness degree r on M. If :_{f R}n __R_{is a differentiable} roughlyB-invexfunction with respect to :_ _{M M}_ __Rn with roughness degree r, at y M , then there exists a function b x y



,



, such that

   

, , T _x

     

, [

 

]

b x y  x y  f y b x y f x f y ,

for each x M such that x y r.

A numerical function f, defined on a B-invex subset M

of Rn_{, is said to be quasi roughly}_B-_invex_{with respect to}

:_{M M} _Rn

   with roughness degree r at y M , if there exists b x



, ,y



:M M 

 

0,1 R, such that

 



 

,



 

f x f y  f y b x y  f y ,

for each x M , and 0  1 such that x y r.

f is said to be quasi roughlyB-invexon M with respect to 



x y,



if it is roughlyB-invexat each y M with respect to the same 

 

x y, .

A _{f R}: n __R_{is quasi roughly}_B-_invex_{with respect to}

η: M × M →Rn_{with roughness degree}_r_{, on} _M __Rn_{, if}

and only if the level set

 



: ,



K  x x M f x 

is B-invex set. A roughly B-invex function, with respect to _:_{M M}_ __Rn_{with roughness degree}_r_{is quasi} roughly B-invex function with respect to same η with roughness degree r. Let _M__Rn_be_B_{-invex set, if}

: n

f R R

:

is differentiable quasi roughlyB-invexwith respect to _ _{M M}_ __Rn_{with roughness degree}_r_{, at}

y M , then there exists a function b x y



,



, such that

   

, , x

 

b x y  x y T f y 0,

A numerical function f, defined on a B-invex subset M

of Rn_{, is said to be pseudo roughly}_B-_invex _{with respect}

to _:_{M M}_ __Rn_{with roughness degree}_r_at _{y M}_ _, if there exists b x



, ,y



:M M 

 

0,1 R, and, there

exists a strictly positive function a: Rn_×_Rn_→_R_{such that}

 



 

,



  

1

  

,

f x  f y f y b x y  f y   a x y

If _{f R}: n __R_{is roughly}_B_{-invex function with} re-spect to :_{M M}_ __Rn

n

 with roughness degree r on

B-invex set M R

n

 , then f is pseudo roughly B-invex function with respect to same ηwith roughness degree r

on M. Let MR be B-invex set and _{f R}: n __R_be a differentiable pseudo roughly B-invex with respect to_:_{M M}_ __Rn_{with roughness degree}_r_{, at}_{y M}_ _, then there exists a function b x y



,



, such that

   

, , x

 

0

   

b x y  x y T f y   f x f y

, for each x M such that x y r.

2. Problem Formulation

Let : n , 1, 2,

j ,

f R R j _ k, and : n ,

i

g R R i = 1, 2, ···, m are real valued roughly B-invex functions on Rn_.

A roughly B-invex multi-objective programming prob-lem is formulated as follows:

(P)

 





, subject to

: 0, 1, 2,

j

n i

Min f x

.

x M  x R g x  i _m

A feasible solution x for (P) is said to be an efficient solution for (P) if and only if there is no other feasible x

for (P) such that, for some i



1, 2, ,_ k



,

 

* _,

 

* _{,for a}

i i j j ll

f x  f x f x  f x j i .

An efficient solution x M for (P) is a properly efficient solution for (P) if there exists a scalar 0

such that for each i i, 1, 2, ,_ k , and each x M

satisfying

 

*

i i

f x  f x , there exists at least one j i

with

 

* ,

j j

f x  f x and

 

*

 

*

 

i i j j

f x f x f x f x 

 _   _ _

    .

Lemma 1 [9]

If : n i

g R 

nR is roughly B-invex with respect to

:M M R

 with roughness degree r, on Rn_,_i_{= 1,}

2, ···, m, then the set

 



n: 0, 1, 2, ,



i

M  x R g x  i _ m

is B-invexset.

Lemma 2 [9]

If : n i

(3)

to _:_{M M}_ __Rn_{with roughness degree}_r_{, on}_Rn_,_i_{= 1,}

2, ···, m, then the set

 



n: 0, 1, 2, ,



i

M  x R g x  i _ m

is B-invexset.

Lemma 3

Let b x y



, ,



:M M 

   

0,1  0,1 . If _{f R}: n __R

is a roughly B-invex function with respect to

:_{M M}_ __Rn

n

 with roughness degree r on a B-invex set MR , then the set

 

x M

A A x



 _ is convex,

where _{A x}

 

_



_{z z R z}_: _ k_, _ _{f x}

 

__{f x}

 

*



_,_{x M}_

1 2 1 2

.

Proof. Let z z, A, then for x x, M and

0  1, we have





   





   

 

   

 



    









 

1 2

1 * 2 *

1 2 *

2 1 2 *

1

(1 )

1

, .

z z

f x f x f x f x

f x f x f x

bf x b f x f x

f x b x x f x

 

 

 

  

 _  _  _ 

   

   

  

 

A

Since f is a roughly B-invex function on a B-invex set

M. Then __z1_{ }



₁ _



_z2_ , and hence A is convex set.

*

For a feasible point x M , we denote _{I x}

 

* _as the index set for binding constraints at x*_, _i.e._,

.

 

*



_:

 

* ₀



i

I x  i g x 

3. Characterizing Efficient Solutions by

Weighting Approach

To characterizing an efficient solution for problem (P) by weighting approach [11] let us scalarize problem (P) to become in the form.

(Pw)

 

1

k j j j

Min w f x





, s.t. x M

j

, where wj0, j1, 2, , , k



kj₁w 1

and fj, j = 1, 2, ···, k are roughly B-invex with respect

to _:_{M M}_ __Rn _{with roughness degree} _r

j on

B-invex set M.

Theorem 1

If x M is an efficient solution for problem (P), then there exist

1

0, 1, 2, , , k 1

j j

w  j _ k



_w_j

such that x is an optimal solution for problem (Pw). Proof. Let x M be an efficient solution for

prob-lem (P), then the system f xj

 

f xj

 

0, j = 1, 2, ···,

k has no solution x M . Upon Lemma 3 and by apply-ing the generalized Gordan theorem [12], there exist

such that 0, 1, 2,

j

p  j _,k

 

x 0, j 1, 2, ,k

   

  

j j

p f x fj ,

and

 

1 1

j j

k k

j j

p p

f x f x

p p

 





.

Denote

1

j

j k

j

p p 





w

0, 1

w  j

, j

then j , 2, , , k



kj₁wj 1, and



kj₁w fj j

 

x 



kj₁w f xj j

 

.

Hence x is an optimal solution for problem (Pw). Theorem 2

If xM is an optimal solution for

 

Pw corre-sponding to wj, then x is an efficient solution for

problem (P) if either one of the following two conditions holds:

(i) wj0, for all j1, 2, , k ; or (ii) x is the unique solution of

 

Pw .

Proof. To proof see V. Chankong, Y. Y. Haimes [11].

4. Characterizing Efficient Solutions by

ε

-Constraint Approach

An ε-constraint approach is one of the common ap-

proaches for characterizing efficient solutions of mul- tiobjective programming problems [11]. In the following we shall characterizing an efficient solution for multiob- jective roughly B-invex programming problem (P) in term of an optimal solution of the following scalar prob-lem.

P ( )_q

 

,

subiect to ,

, 1, 2, , ,

q

j j

Min f x x M 

.

f x  j k j



q

  _ 

where ,fj j1, 2, , k

:

are roughly B-invex with re-spect to _ _{M M}_ __Rn_{with roughness degree}_r

j on

B-invex set M.

Theorem 3

If x M is an efficient solution for problem (P), then x is an optimal solution for problem P ( )q 

cor-responding to j f xj

 

.

Proof.

Let x be not optimal solution for P ( )_q  where

 

, 1, 2, , ,

j f xj j k j q.

   _  So there exists x M such that f xq

 

 f xq

 

,

 

, 1, 2, , ,

j j j

f x   f x j _ k jq.

(4)

is a contradiction. Hence x is an optimal solution for problem P ( )q  .

Theorem 4

Let x M is an optimal solution of P ( )q  for all

q = 1, 2, ···, k, where j f xj

 

, j = 1, 2, ···, k . Then

x is an efficient solution for problem (P).

Proof.

Since x M is an optimal solution for P ( )q  , for

all q = 1, 2, ···, k. So, for each x M , we get

 

,

q q

f x  f x q = 1, 2, ···, k.

This implies that the system f xj

 

f xj

 

0, j = 1,

2, ···, k has no solution x M , i.e. x is an efficient solution for problem (P).

5. Sufficient and Necessary Conditions for

Efficiency

In this section, we discuss the sufficient and necessary conditions for a feasible solution x*_{to be efficient or} properly efficient for problem (P) in the form of the fol-lowing theorems.

Theorem 5

Suppose there exists a feasible solution x*_{for (P), and} scalars _0, _{1, 2, , ,} _0,

 

* , such that

j i

w  j _ k u  i I x

k

 

* *

1

0

j j i i

j i I x

w f x u g x

 

   



* _{. (1)}

If fj, j = 1, 2, ···, k are roughly B-invex with respect

to _:_{M M}_ __Rn



i I

with roughness degree r at x*_and_g

i,

is roughly B-invex with respect to same ηwith roughness degree ri at x*. Then x* is a properly efficient

solution for problem (P).



_x*



Proof.

Since fj, j = 1, 2, ···, k and

 

*

,

i

g i I x are

roughly B-invex with respect to same η, then there exists a function _{b x x}*



_, * such that

k



 

 

   

 

   

 

*

* *

*

* * *

1

* * * *

1

* * * *

( )

* * *

( )

* *

,

, ,

,

, 0.

j j j

j

t k

j j j

t i i i I x

i i i i

i I x i I x

i i i I x

b x x w f x f x

x x b x x w f x

x x b x x u g x

b x x u g x u g x

b x x u g x







 



 _ 

 

 

 _ _

 

  _ _

 

 



 

  





by (1) for each x M such that

 

*

 

*

1 j k i I xmax, j,i

x x r

  

  r



.

Thus, *

 

*

 

*

 

*



* ,

1 1

, k j j , k j j

j j

b x x w f x b x x w f x

 





for all x M , which implies that x*_{is the minimizer of}

 

1

k j j j

f x 





such that

 

* *

1

,

j

j k

j j

w

b x x w







under the constraint g x

 

0 where _{x x}_ * __r _. Hence, from Theorem (4.11) of [11], x*_{is a properly} ef-ficient solution for problem (P).

Theorem 6 Let x* be a feasible solution for (P). If

there exist scalarswj0,j1, 2, , k,

 

* 1

1, 0,

k

j i j

w u i I x



  



,

such that the triplet (x*_,_w

i, ui)satisfies (1) of Theorem (5),

1

k j j j

w f





is strictly roughly B-invex with respect to η: M × M →Rn

with roughness degree r at x*_and_g

i, i I x

 

* is

roughly B-invex with respect to same η with roughness degree ri at x*. Then x* is an efficient solution for

prob-lem (P).

Proof.

Suppose that x*_{is not an efficient solution for (P).} Then, there exists a feasible x M , and index v such that

 

* _,

v v

f x  f x f x

 

 , for all j v .

k

 

*

j f xj

Since 1 j j

j

w f





is strictly roughly B-invex with respect to ηwith rough-ness degree r at x*_{, then there exists a function}_{b x x}*

 

_, * such that

 

   

 

* * *

1 1

* * * *

1

0 ,

0 , ,

k k

j j j j

j j

t k

j j j

b x x w f x w f x

x x b x x w f x



 



 

 _  _

 

 

  _ _



. (2)

Also, roughly B-invexity of gi, with respect

to same ηat x*_{with roughness degree}_r

i implies

 

*

i I x

       

_, * * _, * * * _, *

 

*

i i

x x b x x g x b x x g x g x

 _ _  _ 

 i  ,

i.e.

     

_, * * _, * * _0,



*

i



x x b x x g x i I x

    , (3)

such that _x * max ,

 

i

x r r

(5)

Remark 1

Similarly as in Theorem (5), it can be easily seen that

x*_{becomes properly efficient solution for (P), in the} above theorem, if wj > 0, for all j = 1, 2, ···, k.

Theorem 7

Suppose there exists a feasible solution x*_{for (P), and} scalars wi > 0, j = 1, 2, ···, k, such that

(1)of Theorem (5) holds. If

 

*

0,

i

u  i I x

1

k j j j

w f





is pseudo roughly B-invex with respect to _:_{M M}_ __Rn with roughness degree r at x*_and_g

I is quasi roughly

B-invex with respect to same ηwith roughness degree rIat

x*_{. Then}_x*_{is a properly efficient solution for problem} (P).

Proof.

Since , and gI are quasi

roughly B-invex with respect to ηwith roughness degree

rI at

 

* _0, ₀

I I i

g x g x  u 

*

x , then there exists a function _{b x x}*



_, * such that

t



0

   

 

*

* * * *

, , i i

i I x

x x b x x u g x





_  _

 



,

for all x M such that * _max

i i

x x  r

0



. By using (1), we have

   

* * *

 

*

1

, ,

t k

j j j

x x b x x w f x





_  _

 



which implies

 



* * * * *

1 1

, k j j , k j j

i j

b x x w f x b x x w f x

 





,

since 1

k j j j

w f





is pseudo roughly B-invex with respect to same η with roughness degree r at x*_{which implies that}_x*_{is the} minimizer of

 

1

k j j j

f x 





such that

 

* *

1

,

j

j k

j j

w

b x x w







under the constraint g x

 

0 where _{x x}_ * _≥_max(_r_,

ri). Therefore, x* is a properly efficient solution for

prob-lem (P).

Theorem 8

Suppose that there exist a feasible solution x* _{for (P)} and scalarswj 0,j1, 2, , k,

such that (1)of Theorem (5) holds. Let 1

1

k j j

w

 



, _0,

 

* ,

i

u  i I x

1

k j j j

w f





be strictly pseudo roughly B-invex with respect to

:_{M M} _Rn

   with roughness degree r at _x*_a b ly B-invex with respect to sam η with roughness degree rIat x*. Then x* is an efficient solution

for problem (P).

Proof. Suppose that x* is not an efficient solution for

(P

nd gI

e quasi rough e

).Then, there exists a feasible x for (P), and index v

such that

 

* _,

 

* _{, for all} _.

v v i i

f x  f x f x  f x i r

then there ex

 

* _, * k k *

b x x



w f x



w f x

ists a function _{b x x}*

 

_, * _{such that,}

 

* *

1 1

,

j j j j

j j

b x x

 

 ,

1

k j j j

w f





Strictly pseudo roughly B-invexity of

implies that 0

for all

   

* * *

 

*

1

, ,

t k

j j j

x x b x x w f x





_  _

 



such that

x M _{x x}_ * __r_{. Since}_g

I is quasi

-invex with respect to sam

roughly e η with roughness degree rIat *

B

x and

 

* ₀

I I

g x g x  ,

then _

    

_{x x}_, * * _, * _ *



_

I

b x x g x 0,

for all x M such that

* _max

i i

x x  r.

The proof now similar to the proof o heorem (6). t

x*

ble solution x*_f _{), and} sc

f T

Remark 2

in Theorem (7), it can be easily seen tha Similarly as

becomes properly efficient solution for (P), in the above theorem, if w_j 0,for all 1, 2, ,j _ k.

Theorem 9

e exists a feasi or (P Suppose ther

alars _0, _{1, 2, , ,} _0,

 

*

j i

w  j _ k u  i I x such that

(1)of Th

k

eorem (5) holds. Let 1 j j

j

w f





be pseudo roughly B-invex with respect to

:_{M M} _Rn

   with roughness degree r_I at x*_and

I I

u g be quasi r with

oughly B-invex with respe to same η

roughness degree

ct

I

r at x*_{. Then}_x*_{is a properly} efficient solution for problem (P).

Proof. The proof is similar to the proof of Theorem (7). Theorem 10

there exist a feasible solution x*_{for (P)} Suppose that

an

such that (1)of Theorem (5) holds. If d scalars w_j0,j1, 2, ,_ k,

 

* 1

1, 0,

k

j i j

w u i I x



  



,

 

*

(6)

 

* _.

i I x

k

1 j j

i

w f 



_{Similarly, for}

_{   }

* _, *

i

i I x g x 0, and for 0

small enough is quasi roughly B-invex with respect to _:_{M M}_ __Rn

with roughness degree r at x*_and

I I

u g is strictly to sa

pseudo roughly B-invex with respect me η with roughness degree rIat x*. Then x* is an efficient solution

for problem (P).

Proof. The pr

 

   



* _, * _, * _, *



_0,

 

i *

g x _  x x _b x x  x x  _ i I x_

    .

Thus, for λ sufficiently small and all

 

   

* * *

0,_x _{x x}, _{b x x}, _{x x}, *

  _     _

oof is similar to the proof of Theorem (8

mark 3

in Theorem (7), it can be easily seen that

x*

tion for pr

).

Re is feasible for problem (P). For sufficiently small _{(5) gives}  0,

Similarly as

 

   



* _, * _, * _, *



 

q q

*

f x  _ x x b x x  x x _  f x . (7)

becomes properly efficient solution for (P), in the above theorem, if wj0,for all j1, 2, , k.

Theorem 11 (Necessary Optimality Criteria) Now, for all jq such that Assume that x*_{is a properly efficient solu}

 

   



* _, * _, * _, *



 

j j *

f x  _ x x b x x  x x _  f x (8)

oblem (P). Assume also that there exist a feasible point

x for (P) such that g xi

 

 0,i1, 2, , m, and each

 

*

,

i

g i I x is rough to η: M ×

roughness degree ri at x*. Then,there exists

scalars wj0, j1, 2, , k and

ly B-invex with respect

M →Rn_with

 

*

0,

i

u  i I x ,

such thatthe triplet



*_, _,



j i

x w u satisfies

k

Consider the ratio (see Equation (9))

From (5), .

Simi-larly,







  

* *

, t q 0

N     x x f x 



_,





*

  

t * ₀

j

D    x x f x  ; but, by (8)





,so



,



0

D  , 0 D    .

Thus, the ratio in (9) becomes unbounded, contradicting the proper efficiency of x*_{for (P). Hence, for each}_q_{= 1,} 2, ···, k, the system (5) has no solution. The result then follows from an application of the Farkas Lemma as in [12], namely

 

*

* *

1

0

j j i i

j _{i I x}

w f x u g x

 _

   



. (4)

Proof. Let the following system

l

   

* t *

 

*

* *

1

0 k

j j i i

j _{i I x}

w f x u g x

 _

   



.

   

 

* *

* * *

, 0,

, 0,for al

, 0,

q t

j t

i

x x f x

x x f x j

x x g x i I x



 

  

  

.

q (5)

has a solution for every . Since by the as-*

Theorem 12

Assume that x*_{is an efficient solution for problem (P)} at which the Kuhn-Tucker constraint qualification is sat-isfied. Then, there exist scalars

1, 2, ,

q _ k

n, sumed Slater-type conditio

 

*

1 0, 1, 2, , , k 1

j j

j

w j k w



  _



 ,ui0,i1, 2, , ,

 

0,

i i m

g x g x  i I x ,

*

x

and then from roughly Bi-invexity of gi at with re-spect to η, there exists a function _{b x x}

 

_, *

 su that

   

ch

 

* * * *

, , t

such that

 

*

 

*

1 1

0

k m

j j i i

j i

w f x u g x

 

   



,

 

*

1

0

m i i i

u g x 





.

0, i

b x x  x x g x  i I x . (6)

Therefore from (5) and (6)



*

Proof.

 

_, *

   

_, * _, * t

 

* _0,



i

x x b x x x x g x i I x

  

 _  _ _

     ,

for all

Since every efficient solution is a weak minimum, then by applying Theorem (2.2) of Weir and Mond [13] for x*_, we get, there exists _{w R u R}_ k, _ m such that

 

*

 

* _0,

 

* _0,

t t t

w f x  u g x  u g x 

0

 . Hence for some positive λ small enough

 

   



* _, * _, * _, *



 

* ₀

g x_i _{ }  x x b x x  x x _ g x_i  , _u__0,_w__0,_{w e}t _₀_{, where} _e_



_{1,1, ,1}



__Rk_.











 



 

   



 

   





 

* * * * *

, , ,

,

, _, _, _,

q q

j j

f x f x x x b x x x x

N

D _{f x} _{x x} _{b x x} _{x x} _{f x}

   

   



   

 _ _ _ _ _ 

 

 

 



 _ _ _ _ _ 

 

 

 

 

.

(7)

REFERENCES

[1] B. D. Craven, “Control and Optimization,” Chapman and Hall, London, 1995.

] H. X. Phu, “Six Kind of Roughly Convex Functions,”

Journal of Op plications, Vol. 92,

No. 2, 1997, p [2

timization Theory and Ap

p. 357-375. doi:10.1023/A:1022611314673

[3] S. K. Mishra, S. Y. Wang and K. K. Lai, “General

vexity and Optimization

Invexity over Cones in Vector Optimization,” ized Convexity and Vector Optimization, Nonconvex Optimi- zation and Its Applications,” Springer-Verlag, Berlin, 2009.

[4] S. K. Mishra and G. Giorgi, “In , Nonconvex Optimization and Its Applications,” Springer- Verlag, Berlin, 2008.

[5] S. K. Suneja, S. Khurana and Vani, “Generalized Non- smooth

European Journal of Operational Research, Vol. 186, No.

1, 2008, pp. 28-40. doi:10.1016/j.ejor.2007.01.047 [6] S. Komlosi, T. Rapesak and S. Schaible, “Generalized

tions, Vol. 80, No. 2, 1981, pp. 545-550.

doi:10.1016/0022-247X(81)90123-2 [8] C. R. Bector, S. K. Su

of Preinvex and B-vex Functions,” Journal of Optimiza-

tion Theory and Applications, Vol. 76, No. 3, 1993, pp.

Convexity,” Springer-Verlag, Berlin, 1994.

[7] M. A. Hanson, “On Sufficiency of the Kuhn-Tucker Con-ditions,”Journal of Mathematical Analysis and Applica-

nela and C. Singh, “Generalization

277-287. doi:10.1007/BF00939383

[9] T. Emam, “Roughly B-invex Programming Problems,”

Calcolo, Vol. 48, No. 2, 2011, pp. 173-188.

doi:10.1007/s10092-010-0034-5

[10] T. Morsy, “A Study on Generalized Convex Mathemati- cal Programming Problems, Master Thesis, Faculty of

jective Deci-y,” North-Holland,

Sons, Inc.,

ective Programming,” Bulletin of the

Science,” Suez Canal University, Egypt, 2003. [11] V. Chankong and Y. Y. Haimes, “Multiob

sion Making Theory and Methodolog Amsterdam, 1983.

[12] M. S. Bazaraa and C. M. Shetty, “Nonlinear Program- ming-Theory and Algorithms,” John Wiley and

New York, 1979.

[13] T. Weir and B. Mond, “Generalized Convexity and Dual- ity in Multiple Obj

Australian Mathematical Society, Vol. 39, No. 2, 1989,