Optimality Conditions for Non-Smooth Multi Objective Semi-Infinite Programming
The purpose of this paper is to consider a class of non-smooth multi objective semi-infinite programming problem. Based on the concepts of local cone approximation, K—directional derivative and K—subdifferential, a new generalization of convexity, namely
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Optimality Conditions for Non-Smooth Multi Objective Semi-Infinite Programming Xiaoyan Gao
Abstract The purpose of this paper is to consider a class of non-smooth multi objective semi-infinite programming problem. Based on the concepts of local cone approximation, K—directional derivative and K—subdifferential, a new generalization of convexity, namely generalized uniform K—ðF; a; q; dÞ—convexity, is defined for this problem. For such semi-infinite programming problem, several sufficient optimality conditions are established and proved by utilizing the above defined new classes of functions. The results extend and improve the corresponding results in the literature. Keywords Local cone approximation differential Semi-infinite programming
K—directional derivative K—sub Sufficient optimality condition
58.1 Introduction In recent years, there has been considerable interest in so-called semi-infinite programming problems—the optimization of an objective function in finitely many variables over a feasible region defined by an infinite number of constraints. Semi-infinite programming have been a subject of wide interest since they play a key role in a particular physical or social science situation, i.e., control of robots, mechanical stress of materials, and air pollution abatement etc. To date, many authors investigated the optimality conditions and duality results for semi-infinite programming problems. We can see in [1, 2, 3].
X. Gao (&) School of Science, Xi’an University of Science and Technology, Xi’an 710054, China e-mail: [email protected]
Z. Zhong (ed.), Proceedings of the International Conference on Information Engineering and Applications (IEA) 2012, Lecture Notes in Electrical Engineering 218, DOI: 10.1007/978-1-4471-4847-0_58, Ó Springer-Verlag London 2013
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On the other hand, a large literature was developed around generalized convexity and its applications in mathematical programming. Several authors have extended the basic theoretical results in multi objective programming. The optimality conditions and duality in multi objective programming have not only been used in many theoretical and computational developments in mathematical programming itself but also used in economics, control theory, business problems, and other diverse fields. In particular, Anurag Jayswal [4] obtained the sufficient optimality conditions and duality results under the generalized a unisex type I function. Optimality conditions and generalized Mond-Weir duality for multi objective programming involving n-set functions which satisfy appropriate generalized university V-type-I conditions were formulated in [5]. Kim and Bae [6] formulated no differentiable multi objective programs involving the support functions of a compact convex set. Also, Bae et al. [7] established duality theorems for no differentiable multi objective programming problems under generalized convexity assumptions. Recently, Kim and Lee [8] introduced the no smooth multi objective programming problems involving local Lipchitz functi
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