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eBook details

Title: S-(H,[OMEGA]) Conjugate Duality Theory in Multiobjective Nonlinear Optimization (Report)
Author : Management Science and Engineering
Release Date : January 01, 2008
Genre: Education,Books,Professional & Technical,
Pages : * pages
Size : 87 KB

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1. INTRODUCTION In the duality theory of nonlinear programming problems, conjugate functions play important roles [1,2,8]. So, in order to discuss duality in multiobjective optimization problems, we need to introduce an extended notion of "conjugate" which fits well the multiobjective problems. Tanino and Sawaragi [10] have developed a duality theory in multiobjective convex programming problems under the Pareto optimality criterion. However, their duality theory is not reflexive in the sense analogous to [1, 2]. The nonreflexivity of their duality seems to be caused by the fact that the objective functions in the primal problems are vector-valued, while those in the dual problems are set-valued. If we are concerned with the reflexivity of the duality theory, we will have to start from set-valued objective functions in the primal problems. Then a new problem comes about: what kinds of set-valued functions are suitable for objective functions? Feng [3] extended the results of [10] to a more general framework, and provided a reflexive duality theory in multiobjective optimization based on efficiency. Based on weak efficiency rather than efficiency, Kawasaki [6,7] developed some interesting results by defining conjugate and subgradients via weak supremum. In this article, a new duality theory for weak efficiency is developed with the help of the weak supremum and the generalization of the conjugate relations discussed in [6]. Feng[3] extended Kawasaki's work to a more general case.