Fixed Point Theory and Applications
Volume 2011 (2011), Article ID 175327, 13 pages
doi:10.1155/2011/175327
Research Article

𝜖 -Optimal Solutions in Nonconvex Semi-Infinite Programs with Support Functions

Do Sang Kim1 and Ta Quang Son2

1Department of Applied Mathematics, Pukyong National University, Busan 608-737, Republic of Korea
2Department of Natural Sciences, Nhatrang College of Education, 1 Nguyen Chanh, Nhatrang, Vietnam

Received 6 December 2010; Accepted 29 December 2010

Academic Editor: Jen Chih Yao

Copyright © 2011 Do Sang Kim and Ta Quang Son. 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.

Abstract

Approximate optimality conditions for a class of nonconvex semi-infinite programs involving support functions are given. The objective function and the constraint functions are locally Lipschitz functions on ℝ 𝑛 . By using a Karush-Kuhn-Tucker (KKT) condition, we deduce a necessary optimality condition for local approximate solutions. Then, generalized KKT conditions for the problems are proposed. Based on properties of 𝜖 -semiconvexity and semiconvexity applied to locally Lipschitz functions and generalized KKT conditions, we establish sufficient optimality conditions for another kind of local approximate solutions of the problems. Obtained results in case of nonconvex semi-infinite programs and nonconvex infinite programs are discussed.