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# Symmetric duality for a higher-order nondifferentiable multiobjective programming problem

- Indira P Debnath
^{5}Email author, - Shiv K Gupta
^{5}Email author and - Sumit Kumar
^{6}Email author

**2015**:3

https://doi.org/10.1186/1029-242X-2015-3

© Debnath et al.; licensee Springer. 2015

**Received:**22 September 2014**Accepted:**3 December 2014**Published:**6 January 2015

## Abstract

In this paper, a pair of Wolfe type higher-order nondifferentiable symmetric dual programs over arbitrary cones has been studied and then well-suited duality relations have been established considering *K*-*F* convexity assumptions. An example which satisfies the weak duality relation has also been depicted.

**MSC:**90C29, 90C30, 49N15.

## Keywords

- symmetric duality
- higher-order
*K*-*F*convexity - multiobjective programming
- support function
- efficient solutions

## 1 Introduction

where
be open,
,
, *K*, and *C* are closed convex pointed cones with nonempty interiors in
and
, respectively.

Several researchers have studied the duality relations for different dual problems of (P) under various generalized convexity assumptions. Chen [1] considered a pair of symmetric higher-order Mond-Weir type nondifferentiable multiobjecive programming problems and established duality relations under higher-order *F*-convexity assumptions. Later on, Agarwal *et al.* [2] have filled some of the gap in the work of Chen [1] and proved a strong duality theorem for a Mond-Weir type multiobjective higher-order nondifferentiable symmetric dual program. Khurana [3] considered a pair of Mond-Weir type symmetric dual multiobjective programs over arbitrary cones and established duality results under cone-pseudoinvex and strongly cone-pseudoinvex assumptions. Later on, Kim and Kim [4] extended the results in Khurana [3] to the nondifferentiable multiobjective symmetric dual problem. Gupta and Jayswal [5] studied the higher-order Mond-Weir type multiobjective symmetric duality over cones using higher-order cone-preinvex and cone-pseudoinvex functions, which further extends some of the results in [3, 6, 7].

Agarwal *et al.* [8] formulated a pair of Mond-Weir type nondifferentiable multiobjective higher-order symmetric dual programs over arbitrary cones and established duality theorems under higher-order *K*-*F* convexity assumptions. In the recent work of Suneja and Louhan [9], the authors have considered Wolfe and Mond-Weir type differentiable symmetric higher-order dual pairs. The Mond-Weir type model studied in [9] is similar to the problem considered in Gupta and Jayswal [5]. However, the strong duality result in [9] is for arbitrary cones in
instead of only those cones which contain the nonnegative orthant of
as considered in [5].

In the present paper, a pair of Wolfe type higher-order multiobjective nondifferentiable symmetric dual program have been formulated and we established weak, strong, and converse duality theorems under *K*-*F* convexity assumptions. We also illustrate a nontrivial example of a function which satisfies the weak duality relation.

## 2 Definitions and preliminaries

Let
and
be closed convex cones with nonempty interiors and let
and
be nonempty open sets in
and
, respectively such that
. For a real valued twice differentiable function
defined on
,
denotes the gradient vector of *f* with respect to *x* at
,
denotes the Hessian matrix with respect to *x* at
. Similarly,
,
, and
are also defined.

**Definition 2.1** [8]

**Definition 2.2** [5]

**Definition 2.4**For all , a functional is said to be sublinear with respect to the third variable, if

- (i)
for all ,

- (ii)
, for all and for all .

For convenience, we write .

**Definition 2.5** [8]

*K*-

*F*convex in the first variable at for fixed with respect to

*h*, such that for , , ,

**Definition 2.6** [10]

## 3 Problem formulation

where is fixed.

**Remark 3.1** If
and
, then our problems (WHP) and (WHD) become the problem studied in Suneja and Louhan [9].

Next, we will prove weak, strong, and converse duality results between (WHP) and (WHD).

**Theorem 3.1** (Weak duality)

*Let*

*and*

*be feasible solutions for*(WHP)

*and*(WHD),

*respectively*.

*Assume the following conditions hold*:

- (I)
*is higher*-*order**K*-*F**convex at**u**with respect to**for fixed**v*, - (II)
*is higher*-*order**K*-*G**convex at**y**with respect to**for fixed x*, - (III)
,

*where*

*and*

*are the sublinear functionals with respect to the third variable and satisfy the following conditions*:

*F*in the above expression, we obtain

for .

for .

which contradicts (6). Hence the result. □

**Example 3.1** Let
,
. Let
,
, and
.

Then and . Obviously, .

Now, we shall show that for the primal-dual pair (EP) and (ED), the hypotheses of Theorem 3.1 hold.

**Theorem 3.2** (Strong duality)

*Let*

*be a weak efficient solution of*(WHD).

*Let*

- (I)
*the Hessian matrix**for all**be positive or negative definite*; - (II)
,

*for some**imply that**for all*; - (III)
,

*for all*; - (IV)
*the set of vectors**be linearly independent*; - (V)
, , ,

*for all*.

*Then*

- (I)
*there exists**such that**is feasible for*(WHD)*and* - (II)
*the objective values of*(WHP)*and*(WHD)*are equal*.

*Also*, *if the hypotheses of Theorem *3.1 *are satisfied for all feasible solutions of* (WHP) *and* (WHD), *then*
*is an efficient solution for* (WHD).

*Proof*Since is a weak efficient solution for (WHP), by the Fritz John necessary optimality conditions [11], there exist , , and such that

Using (22) in (18), we have , .

It follows from (11) and (23) that , , which from implies .

Therefore, .

Thus, is a feasible solution for the dual problem.

Now, (15) and (23) yield . Since , .

Again as *E* is a compact convex set in
,
.

Hence, the two objective values are equal.

which contradicts Theorem 3.1. Hence, is the efficient solution of (WHD). □

**Theorem 3.3** (Converse duality)

*Let*

*be a weak efficient solution of*(WHP).

*Let*

- (I)
*the Hessian matrix**for all**be positive or negative definite*; - (II)
,

*for some**implies that**for all*; - (III)
,

*for all*; - (IV)
*the set of vectors**be linearly independent*; - (V)
, , ,

*for all*.

*Then*

- (I)
*there exists**such that**is feasible for*(WHP)*and* - (II)
*the objective values of*(WHP)*and*(WHD)*are equal*.

*Also*, *if the hypotheses of Theorem *3.1 *are satisfied for all feasible solutions of* (WHP) *and* (WHD), *then*
*is an efficient solution for* (WHP).

*Proof* The proof follows along the lines of Theorem 3.2. □

## Declarations

### Acknowledgements

The authors wish to thank the reviewer for her/his valuable and constructive suggestions, which have considerably improved the presentation of the paper. The first author is also grateful to the Ministry of Human Resource and Development, India for financial support to carry out this work.

## Authors’ Affiliations

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