- Research Article
- Open Access

# Efficiency and Generalized Convex Duality for Nondifferentiable Multiobjective Programs

- KwanDeok Bae
^{1}, - YoungMin Kang
^{1}and - DoSang Kim
^{1}Email author

**2010**:930457

https://doi.org/10.1155/2010/930457

© Kwan Deok Bae et al. 2010

**Received:**29 October 2009**Accepted:**15 February 2010**Published:**25 March 2010

## Abstract

We introduce nondifferentiable multiobjective programming problems involving the support function of a compact convex set and linear functions. The concept of (properly) efficient solutions are presented. We formulate Mond-Weir-type and Wolfe-type dual problems and establish weak and strong duality theorems for efficient solutions by using suitable generalized convexity conditions. Some special cases of our duality results are given.

## Keywords

- Support Function
- Constraint Qualification
- Strong Duality
- Duality Result
- Weak Duality

## 1. Introduction and Preliminaries

The concept of efficiency has long played an important role in economics, game theory, statistical decision theory, and in all optimal decision problems with noncomparable criteria. In 1968, Geoffrion [1] proposed a slightly restricted definition of efficiency that eliminates efficient points of a certain anomalous type and lended itself to more satisfactory characterization. He called this new definition proper efficiency. Weir [2] has used proper efficiency to establish some duality results between primal problem and Wolfe type dual problem. He extended the duality results of Wolfe [3] for scalar convex programming problems and some of the more duality results for scalar nonconvex programming problems to vector valued programs.

In 1982, five characterizations of strongly convex sets were introduced by Vial [4]. Based on this, Vial [5] studied a class of functions depending on the sign of the constant . Characteristic properties of this class of sets and related it to strong and weakly convex sets are provided.

Also, Egudo [6] and Weir [2] have used proper efficiency to obtain duality relations between primal problem and Mond-Weir type dual problem. Further, Egudo [7] used the concept of efficiency to formulate duality for multiobjective non-linear programs under generalized convexity assumptions.

Duality theorems for nondifferentiable programming problem with a square root term were obtained by Lal et al. [8]. In 1996, Mond and Schechter [9] studied duality and optimality for nondifferentiable multiobjective programming problems in which each component of the objective function contains the support functions of a compact convex sets. And Kuk et al. [10] defined the concept of invexity for vector-valued functions, which is a generalization of the concept of -invexity concept.

Recently, Yang et al. [11] introduced a class of nondifferentiable multiobjective programming problems involving the support functions of compact convex sets. They established only weak duality theorems for efficient solutions. Subsequently, Kim and Bae [12] formulated nondifferentiable multiobjective programs involving the support functions of a compact convex sets and linear functions.

In this paper, we introduce generalized convex duality for nondifferentiable multiobjective program for efficient solutions. In Section 2 and Section 3, we formulate Mond-Weir type dual and Wolfe type dual problems and establish weak and strong duality under convexity assumptions. In addition, we obtain some special cases of our duality results in Section 4. Our duality results extend and improve well known duality results.

We consider the following multiobjective programming problem:

The functions are assumed to be differentiable. And , for each , is a compact convex set of .

Definition 1.1.

Definition 1.2.

The following definition of -convex function will be used to prove weak duality theorems in Section 2 and Section 3.

If is positive then is said to be strongly convex [4] and if is negative then is said to be weakly convex [5].

In this paper, the proofs of strong duality theorems will invoke the following.

Lemma 1.4.

for each

## 2. Mond-Weir-Type Duality

We introduce a Mond-Weir type dual programming problem and establish weak and strong duality theorems.

Theorem 2.1 (Weak Duality).

Assume that for all feasible for (VOPE) and all feasible for (MVODE), are convex, are -convex and are affine. If also any of the following conditions holds

(a) ;

(b) ,

Proof.

which contradicts (2.1). Also, adding (2.8) and (2.10) and then applying hypothesis (b), we get (2.11). This contradicts to (2.1). Hence (2.3) cannot hold.

It is easy to derive the following result from the corresponding one by Egudo [7].

Corollary 2.2.

Assume that the conclusion of Theorem 2.1 holds between (VOPE) and (MVODE). If is feasible for (MVODE) such that is feasible for (VOPE) and then is efficient for (VOPE) and is efficient for (MVODE).

Theorem 2.3 (Strong Duality).

If be efficient for (VOPE) and assume that satisfies a constraint qualification [14, pages 102-103] for (1.8) for at least one . Then there exist and such that is feasible for (MVODE) and If also weak duality (Theorem 2.1) holds between (VOPE) and (MVODE), then is efficient for (MVODE).

Proof.

we conclude that is feasible for (MVODE). The efficiency of for (MVODE) now follows from Corollary 2.2.

## 3. Wolfe Type Duality

We introduce a Wolfe type dual programming problem and establish weak and strong duality theorems.

Theorem 3.1 (Weak Duality).

Assume that for all feasible for (VOPE) and all feasible for (WVODE), are -convex, are convex and are affine. If also any of the following conditions holds:

(a) ;

(a) ,

Proof.

Now by hypothesis (b), , hence (3.11) implies (3.9), again contradicting (3.1).

The following result can be easily driven from the corresponding one by Egudo [7].

Corollary 3.2.

Assume that the conclusion of Theorem 3.1 holds between (VOPE) and (WVODE). If is feasible for (WVODE) such that is feasible for (VOPE), and then is efficient for (VOPE) and is efficient for (WVODE).

Theorem 3.3 (Strong Duality).

If be efficient for (VOPE) and assume that satisfies a constraint qualification [14, pages 102,103] for (1.8) for at least one . Then there exist and such that is feasible for (WVODE) and , If also weak duality (Theorem 3.1) holds between (VOPE) and (WVODE), then is efficient for (WVODE).

Proof.

we conclude that is feasible for (WVODE).

The efficiency of for (WVODE) now follows from Corollary 3.2.

## 4. Special Cases

We give some special cases of our duality results.

(1)If support functions are excepted and , then our dual programs are reduced to the duals in Egudo [7].

(2)Let . Then and the sets , are compact and convex. If , then (VOPE), (MVODE) and (WVODE) reduce to the corresponding (VP), (VDP) and (VDP) in Lal et al. [8], respectively.

(3)If we replace -convexity by generalized -convexity, then our weak duality theorems reduce to the corresponding ones in Yang et al. [11].

## Authors’ Affiliations

## References

- Geoffrion AM: Proper efficiency and the theory of vector maximization.
*Journal of Mathematical Analysis and Applications*1968, 22: 618–630. 10.1016/0022-247X(68)90201-1MATHMathSciNetView ArticleGoogle Scholar - Weir T: Proper efficiency and duality for vector valued optimization problem.
*Journal of the Australian Mathematical Society, Series A*1987, 43: 21–34. 10.1017/S1446788700028937MATHMathSciNetView ArticleGoogle Scholar - Wolfe P: A duality theorem for nonlinear programming.
*Quarterly of Applied Mathematics*1961, 19: 239–244.MATHMathSciNetGoogle Scholar - Vial, JP: Strong convexity of sets and functions.
*Journal of Mathematical Economics*1982, 9: 187–205. 10.1016/0304-4068(82)90026-XMATHMathSciNetView ArticleGoogle Scholar - Vial JP: Strong and weak convexity of sets and functions.
*Mathematics of Operations Research*1983, 8: 231–259. 10.1287/moor.8.2.231MATHMathSciNetView ArticleGoogle Scholar - Egudo RR: Proper efficiency and multiobjective duality in nonlinear programming.
*Journal of Information and Optimization Sciences*1987, 8: 155–166.MATHMathSciNetView ArticleGoogle Scholar - Egudo RR: Efficiency and generalized convex duality for multiobjective programs.
*Journal of Mathematical Analysis and Applications*1989, 138(1):84–94. 10.1016/0022-247X(89)90321-1MATHMathSciNetView ArticleGoogle Scholar - Lal SN, Nath B, Kumar A: Duality for some nondifferentiable static multiobjective programming problems.
*Journal of Mathematical Analysis and Applications*1994, 186: 862–867. 10.1006/jmaa.1994.1337MATHMathSciNetView ArticleGoogle Scholar - Mond B, Schechter M: Non-differentiable symmetric duality.
*Bulletin of the Australian Mathematical Society*1996, 53: 177–187. 10.1017/S0004972700016890MATHMathSciNetView ArticleGoogle Scholar - Kuk H, Lee GM, Kim DS: Nonsmooth multiobjective programs with V--invexity.
*Indian Journal of Pure & Applied Mathematics*1998, 29: 405–415.MATHMathSciNetGoogle Scholar - Yang XM, Teo KL, Yang XQ: Duality for a class of nondifferentiable multiobjective programming problem.
*Journal of Mathematical Analysis and Applications*2000, 252: 999–1005. 10.1006/jmaa.2000.6991MATHMathSciNetView ArticleGoogle Scholar - Kim DS, Bae KD: Optimality conditions and duality for a class of nondifferentiable multiobjective programming problems.
*Taiwanese Journal of Mathematics*2009, 13(2B):789–804.MATHMathSciNetGoogle Scholar - Chankong V, Haimes YY:
*Multiobjective Decision Making: Theory and Methodology, North-Holland Series in System Science and Engineering*.*Volume 8*. North-Holland, New York, NY, USA; 1983:xvii+406.Google Scholar - Mangasarian OL:
*Nonlinear Programming*. McGraw-Hill, New York, NY, USA; 1969.MATHGoogle Scholar

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