Optimality Conditions and Duality in Nonsmooth Multiobjective Programs
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Research Article Optimality Conditions and Duality in Nonsmooth Multiobjective Programs Do Sang Kim and Hyo Jung Lee Division of Mathematical Sciences, Pukyong National University, Busan 608-737, South Korea Correspondence should be addressed to Do Sang Kim, [email protected] Received 29 October 2009; Accepted 14 March 2010 Academic Editor: Jong Kyu Kim Copyright q 2010 D. S. Kim and H. J. Lee. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We study nonsmooth multiobjective programming problems involving locally Lipschitz functions and support functions. Two types of Karush-Kuhn-Tucker optimality conditions with support functions are introduced. Sufficient optimality conditions are presented by using generalized convexity and certain regularity conditions. We formulate Wolfe-type dual and Mond-Weirtype dual problems for our nonsmooth multiobjective problems and establish duality theorems for weak Pareto-optimal solutions under generalized convexity assumptions and regularity conditions.
1. Introduction Multiobjective programming problems arise when more than one objective function is to be optimized over a given feasible region. Pareto optimum is the optimality concept that appears to be the natural extension of the optimization of a single objective to the consideration of multiple objectives. In 1961, Wolfe 1 obtained a duality theorem for differentiable convex programming. Afterwards, a number of different duals distinct from the Wolfe dual are proposed for the nonlinear programs by Mond and Weir 2. Duality relations for multiobjective programming problems with generalized convexity conditions were given by several authors 3–10. Majumdar 11 gave sufficient optimality conditions for differentiable multiobjective programming which modified those given in Singh 12 under the assumption of convexity, pseudoconvexity, and quasiconvexity of the functions involved at the Pareto-optimal solution. Subsequently, Kim et al. 13 gave a counterexample showing that some theorems of Majumdar 11 are incorrect and establish sufficient optimality theorems for weak Paretooptimal solutions by using modified conditions. Later on, Kim and Schaible 6 introduced
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Journal of Inequalities and Applications
nonsmooth multiobjective programming problems involving locally Lipschitz functions for inequality and equality constraints. They extended sufficient optimality conditions in Kim et al. 13 to the nonsmooth case and established duality theorems for nonsmooth multiobjective programming problems involving locally Lipschitz functions. In this paper, we apply the results in Kim and Schaible 6 for this problem to nonsmooth multiobjective programming problem involving support functions. We introduce nonsmooth multiobjective programming problems involving locally Lipschitz functions and support functions for inequality and equality constraints. Two kinds of sufficie
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