Efficiency and Generalized Convex Duality for Nondifferentiable Multiobjective Programs

Kwan Deok Bae,Young Min Kang,Do Sang Kim

doi:10.1155/2010/930457

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.

Highlights

Introduction and PreliminariesThe 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 1982, five characterizations of strongly convex sets were introduced by Vial 4
Vial 5 studied a class of functions depending on the sign of the constant ρ

Summary

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. Weir 2 has used proper efficiency to establish some duality results between primal problem and Wolfe type dual problem. 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. 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. Kim and Bae 12 formulated nondifferentiable multiobjective programs involving the support functions of a compact convex sets and linear functions. We introduce generalized convex duality for nondifferentiable multiobjective program for efficient solutions. Lemma 1.4. (Chankong and Haimes [13, Theorem 4.1]) x0 is an efficient solution for VOPE if and only if x0 solves the following: Pk 0 minimize fk x s x | Ck subject to fi x s x | Ci fi x0 s x0 | Ci , ∀i / k, 1.8 gj x 0, j 1, . . . , m, hl x 0, l 1, . . . , q for each k 1, . . . , p

Mond-Weir-Type Duality

Wolfe Type Duality