# Second order symmetric duality in nondifferentiable multiobjective programming.

**ABSTRACT** A pair of Mond–Weir type second order symmetric nondifferentiable multiobjective programs is formulated. Weak, strong and converse duality theorems are established under η-pseudobonvexity assumptions. Special cases are discussed to show that this paper extends some work appeared in this area.

**0**Bookmarks

**·**

**71**Views

- [Show abstract] [Hide abstract]

**ABSTRACT:**In this paper, a pair of Mond–Weir type nondifferentiable multiobjective second-order symmetric dual programs over arbitrary cones is formulated. Weak, strong and converse duality theorems are established under second-order K–F-convexity/K–η-bonvexity assumptions. A self duality theorem is also obtained by assuming the functions involved to be skew-symmetric.Optimization Letters 01/2010; 4:293-309. · 1.65 Impact Factor - SourceAvailable from: Navdeep Kailey[Show abstract] [Hide abstract]

**ABSTRACT:**In this paper, a pair of Wolfe type higher-order symmetric nondifferentiable multiobjective programs over arbitrary cones is formulated and appropriate duality relations are then established under higher-order-K-(F, alpha, rho, d)-convexity assumptions. We also illustrate an example which is higher-order K-(F, alpha, rho, d)-convex but not higher-order K-F-convex. Special cases are also discussed to show that this paper extends some of the known work appeared in the literature.Journal of Inequalities and Applications 12/2012; · 0.82 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**A class of multi-objective fractional programming problems (MFP) are considered where the involved functions are locally Lipschitz. In order to deduce our main results, we give the definition of the generalized (F,θ,ρ,d)-convex class about the Clarke’s generalized gradient. Under the above generalized convexity assumption, necessary and sufficient conditions for optimality are given. Finally, a dual problem corresponding to (MFP) is formulated, appropriate dual theorems are proved.Journal of Global Optimization 01/2007; 38(4):653-666. · 1.31 Impact Factor

Page 1

Second order symmetric duality

in nondifferentiable multiobjective

programming

Izhar Ahmad

Department of Mathematics, Aligarh Muslim University, Aligarh 202 002, India

Received 22 August 2003; received in revised form 25 May 2004; accepted 5 June 2004

Abstract

A pair of Mond–Weir type second order symmetric nondifferentiable multiobjective

programs is formulated. Weak, strong and converse duality theorems are established

under g-pseudobonvexity assumptions. Special cases are discussed to show that this

paper extends some work appeared in this area.

? 2004 Elsevier Inc. All rights reserved.

Keywords: Nondifferentiable programming; Multiobjective symmetric duality; g-Pseudobonvexity;

Efficient solutions; Properly efficient solutions

1. Introduction

Symmetric duality in mathematical programming in which the dual of the

dual is the primal problem was first introduced by Dorn [6]. Subsequently,

Dantzig et al. [5], Mond [13] and Bazaraa and Goode [1] formulated a pair

of symmetric dual programs and established duality under convexity–concavity

0020-0255/$ - see front matter ? 2004 Elsevier Inc. All rights reserved.

doi:10.1016/j.ins.2004.06.002

E-mail address: iahmad@postmark.net

Information Sciences 173 (2005) 23–34

www.elsevier.com/locate/ins

Page 2

assumptions. Later, Mond and Weir [16] presented a distinct pair of symmetric

dual programs which allows the weakening of convexity–concavity conditions

to pseudoconvexity–pseudoconcavity.

Weir and Mond [19] discussed symmetric duality in multiobjective program-

ming by using the concept of efficiency. Chandra and Prasad [4] presented a

pair of multiobjective programming problems by associating a vector valued

infinite game to this pair. Gulati et al. [9] also established duality results for

multiobjective symmetric dual problems without nonnegativity constraints.

Mangasarian [11] considered a nonlinear program and discussed second

order duality under certain inequalities. Mond [14] assumed rather simple in-

equalities. Bector and Chandra [2] defined the functions satisfying the inequal-

ities in [14] to be bonvex–boncave. To give examples of bonvex–boncave

functions, Mond [14] has shown that a convex (concave) quadratic or linear

function is bonvex (boncave). Mangasarian [11, p. 609] and Mond [14, p. 93]

have also indicated possible computational advantages of the second order

duals over the first order duals. Yang [20] also discussed second order Mangas-

arian type dual formulation under generalized representation conditions.

Gulati et al. [8] studied two distinct pairs of second order symmetric dual

programs under g-bonvexity and g-pseudobonvexity. Recently, Hou and Yang

[10] and Yang et al. [21] generalized the results in [8] to nondifferentiable pro-

grams on the lines of [15] involving second order F-convex and second order

F-pseudoconvex functions.

In this paper, we formulate a new pair of second order symmetric nondiffer-

entiable multiobjective dual programs of Mond–Weir type and prove duality

theorems under g-pseudobonvexity assumptions. These results include, as spe-

cial cases, recent duality results for multiobjective symmetric programs given

by Suneja et al. [18] and for single objective symmetric programs studied by

Mond and Schechter [15], Gulati et al. [8,9], Mishra [12], and Hou and Yang

[10].

2. Preliminaries

Let F be a twice differentiable real valued function of x and y, where x2Rn

and y2Rm. Then $xF and $yF denote the gradient vectors with respect to x

and y respectively. $xxF and $yyF are respectively, the n·n and m·m symmet-

ric Hessian matrices. ($xxFr)y denotes the matrix whose (i,j)th element is

o

oyiðrxxFrÞj, where r2Rn.

The following conventions for vectors in Rnwill be used:

x=u ( ) xi=ui; i ¼ 1;2;...;n;

x ? u ( ) xi=ui; i ¼ 1;2;...;n; and x 6¼ u;

x > u ( ) xi> ui; i ¼ 1;2;...;n:

24

I. Ahmad / Information Sciences 173 (2005) 23–34

Page 3

Definition 1. Let C be a compact convex set in Rn. The support function s(xjC)

of C is defined by

sðxjCÞ ¼ maxfxty : y 2 Cg:

Definition 2. Let D be a nonempty convex set in Rn, and let w:D!R be con-

vex. Then z is called a subgradient of w at ? x 2 D if

wðxÞ=wð? xÞ þ ztðx ?? xÞ

for all x 2 D:

A support function s(xjC), being convex and everywhere finite, has a subdif-

ferential, that is, there exists z such that s(yjC)=s(xjC)+zt(y?x) for all x2C.

The set of all subdifferentials of s(xjC) is given by

osðxjCÞ ¼ fz 2 C : ztx ¼ sðxjCÞg:

For a set S, the normal cone to S at a point x2S is defined by

NSðxÞ ¼ fy : ytðz ? xÞ50

When C is a compact convex set, then y is in NC(x) if and only if s(yjC)=xty,

i.e., x is a subdifferential of s at y.

Consider the following multiobjective programming problem:

fðxÞ ¼ ½f1ðxÞ;f2ðxÞ;...;fkðxÞ?

subject to

x 2 X ¼ fx 2 Rn: gðxÞ50g;

where f:Rn!Rkand g:Rn!Rm.

for all z 2 Sg:

ðPÞ

Minimize

Definition 3 [7]. A point ? x 2 X is said to be an efficient solution of (P), if there

exists no other x2X such that fðxÞ6fð? xÞ.

A point ? x is said to be properly efficient solution of (P), if it is efficient and if

there exists a scalar M>0 such that, for each i2{1,2,...,k} and x2X satisfying

fiðxÞ < fið? xÞ, we have

fið? xÞ ? fiðxÞ

fjðxÞ ? fjð? xÞ5M;

for some j such that fjðxÞ > fjð? xÞ.

Definition 4 [3]. A point ? x 2 X is said to be a weak efficient solution of (P), if

there exists no other x2X with fðxÞ < fð? xÞ.

It readily follows that if ? x 2 X is efficient, then it is also weak efficient.

Definition 5. A real twice differentiable function F(x,y) defined on X·Y, where

X and Y are open sets in Rnand Rmrespectively, is said to be g-pseudobonvex

I. Ahmad / Information Sciences 173 (2005) 23–34

25

Page 4

at u2X for fixed v2Y, if there exists a function g:X·X!Rnsuch that for

r2Rn, x2X,

gtðx;uÞ½rxFðu;vÞ þ rxxFðu;vÞr?=0 ) Fðx;vÞ=Fðu;vÞ ?1

2rtrxxFðu;vÞr:

A real twice differentiable function F(x,y):X·Y!R is said to be g-pseudo-

boncave if ?F is g-pseudobonvex.

3. Mond–Weir type symmetric duality

We now state the following pair of second order nondifferentiable multiob-

jective symmetric programs and establish weak, strong and converse duality

theorems.

Primal ðMPÞ:

Minimize Lðx;y;z;pÞ ¼ L1ðx;y;z1;p1Þ;L2ðx;y;z2;p2Þ;...;Lkðx;y;zk;pkÞ

subject to

½?

X

i¼1

k

kiryfiðx;yÞ ? ziþ ryyfiðx;yÞpi

??50;

ð1Þ

ytX

i¼1

k > 0;

zi2 Di; i ¼ 1;2;...;k:

k

ki ryfiðx;yÞ ? ziþ ryyfiðx;yÞpi

??=0;

ð2Þ

ð3Þ

ð4Þ

Dual ðMDÞ:

Maximize Hðu;v;w;rÞ ¼ H1ðu;v;w1;r1Þ;H2ðu;v;w2;r2Þ;...;Hkðu;v;wk;rkÞ

subject to

½?

X

i¼1

k

kirxfiðu;vÞ þ wiþ rxxfiðu;vÞri

½?=0;

ð5Þ

utX

i¼1

k > 0;

wi2 Ci; i ¼ 1;2;...;k;

k

kirxfiðu;vÞ þ wiþ rxxfiðu;vÞri

½?50;

ð6Þ

ð7Þ

ð8Þ

where

Liðx;y;zi;piÞ ¼ fiðx;yÞ þ sðxjCiÞ ? ytzi?1

2pt

iryyfiðx;yÞpi;

26

I. Ahmad / Information Sciences 173 (2005) 23–34

Page 5

Hiðu;v;wi;riÞ ¼ fiðu;vÞ ? sðvjDiÞ þ utwi?1

ki2R, pi2Rm, ri2Rn, i=1,2,...,k, and fi, i=1,2,...,k are thrice differentiable

functions from Rn·Rmto R, Ciand Di, i=1,2,...,k are compact convex sets in

Rnand Rm. Also we take p=(p1,p2,...,pk), r=(r1,r2,...,rk), w=(w1,w2,...,wk)

and z=(z1,z2,...,zk).

2rt

irxxfiðu;vÞri;

Theorem 1 (Weak duality). Let (x,y,k,z,p) be feasible for (MP) and

(u,v,k,w,r) be feasible for (MD). Let

(i)

(ii)

(iii) g1(x,u)+u=0, and

(iv) g2(v,y)+y=0.

Pk

Pk

i¼1ki½fið?;vÞ þ ð?Þtwi? be g1-pseudobonvex at u,

i¼1ki½fiðx;?Þ ? ð?Þtzi? be g2-pseudoboncave at y,

Then

Lðx;y;z;pÞiHðu;v;w;rÞ:

Proof. From (5) and hypothesis (iii), we have

gt

1ðx;uÞ

X

i¼1

k

kirxfiðu;vÞ þ wiþ rxxfiðu;vÞri

½?

= ? utX

k

i¼1

kirxfiðu;vÞ þ wiþ rxxfiðu;vÞri

½?=0

ðby (6)Þ:

Therefore, hypothesis (i) implies

X

i¼1

k

kifiðx;vÞ þ xtwi

½?=

X

i¼1

k

ki fiðu;vÞ þ utwi?1

2rt

irxxfiðu;vÞri

??

:

ð9Þ

From (1) and hypothesis (iv), it follows that

gt

2ðv;yÞ

X

i¼1

k

kiryfiðx;yÞ ? ziþ ryyfiðx;yÞpi

??

5 ? ytX

k

i¼1

ki ryfiðx;yÞ ? ziþ ryyfiðx;yÞpi

??50

ðusing (2)Þ;

which, in view of hypothesis (ii) gives

X

i¼1

k

kifðx;vÞ ? vtzi

½?5

X

i¼1

k

ki fiðx;yÞ ? ytzi?1

2pt

iryyfiðx;yÞpi

??

:

ð10Þ

I. Ahmad / Information Sciences 173 (2005) 23–34

27

Page 6

The relations (9) and (10) yield

Xk

i¼1ki fiðx;yÞ þ xtwi? ytzi?1

k

ki fiðu;vÞ þ utwi? vtzi?1

2pt

iryyfiðx;yÞpi

??

=

X

i¼1

2rt

irxxfðu;vÞri

??

:

Finally, using xtwi5s(xjCi), wi2Ci, i=1,2,...,k and vtzi5s(vjDi), zi2Di,

i=1,2,...,k, we obtain

Xk

i¼1ki fiðx;yÞ þ sðxjCiÞ ? ytzi?1

k

ki fiðu;vÞ ? sðvjDiÞ þ utwi?1

2pt

iryyfiðx;yÞpi

??

=

X

i¼1

2rt

irxxfðu;vÞri

??

:

Hence

Lðx;y;z;pÞiHðu;v;w;rÞ:

?

Theorem 2 (Strong duality). Let f be thrice differentiable on Rn·Rm. Let

ð? x;? y;?k;? z;? pÞ be a weak efficient solution for (MP); fix k ¼?k in (MD) and suppose

that

(A1) $yy fiis nonsingular for all i=1,2,...,k,

(A2) the matrixPk

(A3) the set ryf1?? z1þryyf1? p1;ryf2?? z2þryyf2? p2;...;ryfk?? zkþryyfk? pk

is linearly independent,

i¼1?kiðryyfi? piÞyis positive or negative definite, and

??

then there exist ? wi2 Rn, i=1,2,...,k such that ? p ¼ 0, ð? x;? y;?k; ? w;? r ¼ 0Þ is feasible

for (MD) and

Lð? x;? y;? z;? pÞ ¼ Hð? x;? y; ? w;? rÞ:

Also, if the hypotheses of Theorem 1 are satisfied for all feasible solutions of

(MP) and (MD), then ð? x;? y;?k; ? w;? rÞ is a properly efficient solution for (MD).

Proof. Since ð? x;? y;?k;? z;? pÞ is a weak efficient solution of (MP), by the Fritz–John

conditions [17], there exist a2Rk, b2Rm, m2R, d2Rkand ? wi2 Rn, i=1,2,...,k

such that

X

i¼1

k

ai rxfiþ ? wi?1

2ðryyfi? piÞx? pi

??

þ

X

i¼1

k

?ki ryxfiþ ryyfi? piÞx

??ðb ? m? yÞ ¼ 0;

?

ð11Þ

28

I. Ahmad / Information Sciences 173 (2005) 23–34

Page 7

? wi2 Ci;

? xt? wi¼ sð? xjCiÞ;

i ¼ 1;2;...;k;

ð12Þ

Xk

i¼1ðai? m?kiÞ½ryfi?? zi? þ

X

i¼1

k

?ki½ryyfi?ðb ? m? y ? m? piÞ

þ

X

i¼1

k

ðryyfi? piÞyðb ? m? yÞ?ki?1

2ai? pi

??

¼ 0;

ð13Þ

ðb ? m? yÞtryfi?? ziþ ryyfi? pi

½ðb ? m? yÞ?ki? ai? pi?tryyfi;

ai? y þ?kiðb ? m? yÞ 2 NDið? ziÞ;

btXk

k

?kiðryfi?? ziþ ryyfi? piÞ ¼ 0;

??? di¼ 0;

i ¼ 1;2;...;k;

i ¼ 1;2;...;k;

i ¼ 1;2;...;k;

ð14Þ

ð15Þ

ð16Þ

i¼1?kiðryfi?? ziþ ryyfi? piÞ ¼ 0;

ð17Þ

m? ytX

i¼1

ð18Þ

dt?k ¼ 0;

ða;b;m;dÞ=0;

ða;b;m;dÞ 6¼ 0:

In view of?k > 0 and d=0, it readily follows from (19) that d=0. Therefore

from (14), we have

ðb ? m? yÞtryfi?? ziþ ryyfi? pi

Since $yyfiis nonsingular for i=1,2,...,k, (15) yields

ðb ? m? yÞ?ki¼ ai? pi;

Now from (13),

ð19Þ

ð20Þ

ð21Þ

??¼ 0;

i ¼ 1;2;...;k:

ð22Þ

i ¼ 1;2;...;k:

ð23Þ

X

i¼1

k

ðai? m?kiÞðryfi?? ziþ ryyfi? piÞ þ1

2

X

i¼1

k

?kiðryyfi? piÞyðb ? m? yÞ ¼ 0:

ð24Þ

On multiplying (24) by ðb ? m? yÞtfrom the left and using (22), we get

ðb ? m? yÞtX

i¼1

which by the hypothesis (A2) implies

k

?kiðryyfi? piÞyðb ? m? yÞ ¼ 0;

b ¼ m? y:

ð25Þ

I. Ahmad / Information Sciences 173 (2005) 23–34

29

Page 8

Therefore, from (24)

X

i¼1

k

ðai? m?kiÞðryfi?? ziþ ryyfi? piÞ ¼ 0;

which by the hypothesis (A3) yields

ai¼ m?ki;

Suppose m=0. Then from (25) and (26), we get b=0 and a=0 respectively.

Thus (a,b,m,d)=0, a contradiction to (21). Hence

i ¼ 1;2;...;k:

ð26Þ

m > 0:

ð27Þ

Since?ki> 0, i=1,2,...,k, from (26) and (27), we get

i ¼ 1;2;...;k:

Using (25) in (23), we have

ai> 0;

ai? pi¼ 0;

i ¼ 1;2;...;k;

and hence

? pi¼ 0;

i ¼ 1;2;...;k:

ð28Þ

Now using relations (25) and (28) in (11), it follows that

X

i¼1

k

ai½rxfiþ ? wi? ¼ 0;

which by (26) gives

X

i¼1

k

?ki½rxfiþ ? wi? ¼ 0;

and hence, we also have

? xtX

k

i¼1

?ki½rxfiþ ? wi? ¼ 0:

Therefore ð? x;? y;?k; ? w;? r ¼ 0Þ is feasible for (MD).

From (16) and (25), we obtain

? yt? zi¼ sð? yjDiÞ;

By (12) and (29), we get

i ¼ 1;2;...;k:

ð29Þ

fið? x;? yÞ þ sð? xjCiÞ ? ? yt? zi?1

¼ fið? x;? yÞ ? sð? yjDiÞ þ? xt? wi?1

Thus ð? x;? y;?k; ? w;? r ¼ 0Þ is a feasible solution of the dual problem (MD) and

Lð? x;? y;? z;? pÞ ¼ Hð? x;? y; ? w;? rÞ:

2? pt

iryyfið? x;? yÞ? pi

2? rt

irxxfið? x;? yÞ? ri;

i ¼ 1;2;...;k:

ð30Þ

30

I. Ahmad / Information Sciences 173 (2005) 23–34

Page 9

Now we show proper efficiency of ð? x;? y;?k; ? w;? rÞ for (MD) by exhibiting a con-

tradiction. If ð? x;? y;?k; ? w;? rÞ is not efficient for (MD) then there exists a feasible

solution ð? u;? v;?k; ? w;? rÞ for (MD) such that

Hð? x;? y; ? w;? rÞ6Hð? u;? v; ? w;? rÞ:

In view of (30), it follows that

Lð? x;? y;? z;? pÞ6Hð? u;? v; ? w;? rÞ;

which contradicts Theorem 1. If ð? x;? y;?k; ? w;? rÞ is not properly efficient for (MD),

then for some feasible ð? u;? v;?k; ? w;? rÞ of (MD) and for some i,

fið? u;? vÞ ? sð? vjDiÞ þ ? ut? wi?1

> fið? x;? yÞ ? sð? yjDiÞ þ? xt? wi?1

and

fið? u;? vÞ ? sð? vjDiÞ þ ? ut? wi?1

? fið? x;? yÞ ? sð? yjDiÞ þ? xt? wi?1

fjð? x;? yÞ ? sð? yjDjÞ þ? xt? wj?1

fjð? u;? vÞ ? sð? vjDjÞ þ ? ut? wj?1

2? rt

irxxfið? u;? vÞ? ri

2? rt

irxxfið? x;? yÞ? ri

2? rt

irxxfið? u;? vÞ? ri

??

2? rt

irxxfið? x;? yÞ? ri

??

> M

2? rt

jrxxfjð? x;? yÞ? rj

???

?

?

2? rt

jrxxfjð? u;? vÞ? rj

??

for any M>0, and all j satisfying

fjð? x;? yÞ ? sð? yjDjÞ þ? xt? wj?1

> fjð? u;? vÞ ? sð? vjDjÞ þ ? ut? wj?1

? xt? wi¼ sð? xjCiÞ and ? yt? zi¼ sð? yjDiÞ, i=1,2,...,k.

This means that

fið? u;? vÞ ? sð? vjDiÞ þ ? ut? wi?1

?

can be made arbitrarily large. Thus for any?k > 0,

k

?ki fið? u;? vÞ ? sð? vjDiÞ þ ? ut? wi?1

2? rt

jrxxfjð? x;? yÞ? rj

2? rjrxxfjð? u;? vÞ? rj;

2? rt

irxxfið? u;? vÞ? ri

??

? fið? x;? yÞ þ sð? xjCiÞ ? ? yt? zi?1

2? rt

irxxfið? x;? yÞ? ri

?

X

i¼1

2? rt

irxxfið? u;? vÞ? ri

??

>

X

i¼1

k

?ki fið? x;? yÞ þ sð? xjCiÞ ? ? yt? zi?1

2? rt

irxxfið? x;? yÞ? ri

??

:

This again contradicts Theorem 1.

h

I. Ahmad / Information Sciences 173 (2005) 23–34

31

Page 10

A converse duality theorem may be merely stated as its proof would run

analogously to that of Theorem 2.

Theorem 3 (Converse duality). Let f be thrice differentiable on Rn·Rm. Let

ð? u;? v;?k; ? w;? rÞ be a weak efficient solution for (MD); fix k ¼?k in (MP) and suppose

that

(B1)

(B2)

(B3)

$xxfiis nonsingular for all i=1,2,...,k,

the matrixPk

the set rxf1? ? w1þrxxf1? r1;rxf2? ? w2þrxxf2? r2;...;rxfk? ? wkþrxxfk? rk

is linearly independent,

i¼1?kiðrxxfi? riÞxis positive or negative definite, and

ðÞ

then there exist ? zi2 Rm, i=1,2,...,k such that ? r ¼ 0, ð? u;? v;?k;? z;? p ¼ 0Þ is feasible

for (MP) and

Lð? u;? v;? z;? pÞ ¼ Hð? u;? v; ? w;? rÞ:

Also, if the hypotheses of Theorem 1 are satisfied for all feasible solutions

of (MP) and (MD), then ð? u;? v;?k;? z;? pÞ is a properly efficient solution for

(MP).

4. Special cases

(i) Let Ci={0} and Di={0}, i=1,2,...,k. Then (MP) and (MD) are reduced

to the second order multiobjective symmetric dual programs of Suneja

et al. [18]. If in addition p=0, r=0. Then we get the multiobjective sym-

metric dual pair of Gulati et al. [9].

(ii) If k=1 in (MP) and (MD), then we obtain nondifferentiable symmetric

dual programs studied by Hou and Yang [10].

(iii) If in (MP) and (MD), k=1, Ci={0} and Di={0}, i=1,2,...,k, then we

get the symmetric dual programs of Gulati et al. [8] and Mishra [12] with

the addition of nonnegativity constraints x=0 and y=0 in (MP) and

(MD) respectively.

(iv) If k=1, p=0 and r=0, then we obtain symmetric dual multiobjective pro-

gramming problems studied by Mond and Schechter [15].

(v) From the symmetry of primal and dual problems (MP) and (MD), we can

construct other new symmetric dual pairs. For example, if we take

Ci={Aiy:ytAiy51 and Di={Bix:xtBix51, i=1,2,...,k, where Aiand

Bi, i=1,2,...,k are positive semidefinite matrices, then it can be easily

verified that ðxtAixÞ

a number of new symmetric dual pairs and duality results can be

obtained.

1

2¼ sðxjCiÞ and ðytBiyÞ

1

2¼ sðyjDiÞ, i=1,2,...,k. Thus,

32

I. Ahmad / Information Sciences 173 (2005) 23–34

Page 11

5. Conclusion

In this article, a new pair of Mond–Weir type nondifferentiable multiobjec-

tive second order symmetric dual programs is presented and duality relations

between primal and dual problems are established. The nondifferentiability

terms in the form of support functions have been included in the objective

functions of each problem. The results developed in this paper improve and

generalize a number of existing results in the literature. These results can be

further generalized for minimax mixed integer programs, wherein some of

the primal and dual variables are constrained to belong to some arbitrary sets

e.g., the sets of integers.

Acknowledgements

The author wish to thank the referees for their valuable suggestions which

have improved the presentation of this paper. The author is also thankful to

one of the referees to draw his attention towards some similar results studied

by Yang et al. [22].

References

[1] M.S. Bazaraa, J.J. Goode, On symmetric duality in nonlinear programming, Operations

Research 21 (1973) 1–9.

[2] C.R. Bector, S. Chandra, Generalized bonvexity and higher order duality for fractional

programming, Opsearch 24 (1987) 143–154.

[3] J.M. Borwein, Optimization with respect to partial ordering, D.Phil. Thesis, Oxford

University, 1974.

[4] S. Chandra, D. Prasad, Symmetric duality in multiobjective programming, Journal of the

Australian Mathematical Society 35 (1993) 198–206.

[5] G.B. Dantzig, E. Eisenberg, R.W. Cottle, Symmetric dual nonlinear programs, Pacific Journal

of Mathematics 15 (1965) 809–812.

[6] W.S. Dorn, Self-dual quadratic programs, SIAM Journal on Applied Mathematics 9 (1961)

51–54.

[7] A.M. Geoffrion, Proper efficiency and the theory of vector optimization, Journal of

Mathematical Analysis and Applications 22 (1968) 618–630.

[8] T.R. Gulati, I. Ahmad, I. Husain, Second order symmetric duality with generalized convexity,

Opsearch 38 (2001) 210–222.

[9] T.R. Gulati, I. Husain, A. Ahmed, Multiobjective symmetric duality with invexity, Bulletin of

the Australian Mathematical Society 56 (1997) 25–36.

[10] S.H. Hou, X.M. Yang, On second order symmetric duality in nondifferentiable programming,

Journal of Mathematical Analysis and Applications 255 (2001) 491–498.

[11] O.L. Mangasarian, Second and higher order duality in nonlinear programming, Journal of

Mathematical Analysis and Applications 51 (1975) 607–620.

[12] S.K. Mishra, Second order symmetric duality in mathematical programming with F-convexity,

European Journal of Operational Research 127 (2000) 507–518.

I. Ahmad / Information Sciences 173 (2005) 23–34

33

Page 12

[13] B. Mond, A symmetric dual theorem for nonlinear programs, Quarterly Journal of Applied

Mathematics 23 (1965) 265–269.

[14] B. Mond, Second order duality for nonlinear programs, Opsearch 11 (1974) 90–99.

[15] B. Mond, M. Schechter, Nondifferentiable symmetric duality, Bulletin of the Australian

Mathematical Society 53 (1996) 177–188.

[16] B. Mond, T. Weir, Generalized Concavity and Duality, in: S. Schaible, W.T. Ziemba (Eds.),

Generalized Concavity and Duality in Optimization and Economics, Academic Press, New

York, 1981, pp. 263–279.

[17] M. Schechter, More on subgradient duality, Journal of Mathematical Analysis and

Applications 71 (1979) 251–262.

[18] S.K. Suneja, C.S. Lalitha, S. Khurana, Second order symmetric duality in multiobjective

programming, European Journal of Operational Research 144 (2003) 492–500.

[19] T. Weir, B. Mond, Symmetric and self duality in multiobjective programming, Asia-Pacific

Journal of Operational Research 5 (1988) 124–133.

[20] X.Q. Yang, Second order global optimality conditions for convex composite optimization,

Mathematical Programming 81 (1998) 327–347.

[21] X.M. Yang, X.Q. Yang, K.L. Teo, Nondifferentiable second order symmetric duality in

mathematical programming with F-convexity, European Journal of Operational Research 144

(2003) 554–559.

[22] X.M. Yang, X.Q. Yang, K.L. Teo, S.H. Hou, Second order symmetric duality in

nondifferentiable multiobjective programming with F-convexity, European Journal of Oper-

ational Research, in press.

34

I. Ahmad / Information Sciences 173 (2005) 23–34