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Necessary and sufficient conditions for ε -optimal solutions of convex infinite programming problems are established. These Kuhn-Tucker type conditions are derived based on a new version of Farkas' lemma proposed recently. Conditions for ε -duality and ε -saddle points are also given. Keywords: ε -solution, ε -duality, ε -saddle point. | TẠP CHÍ PHÁT TRIỂN KH CN TẬP 10 SỐ 12 - 2007 APPROXIMATE OPTIMALITY CONDITIONS AND DUALITY FOR CONVEX INFINITE PROGRAMMING PROBLEMS Nguyen Dinh 1 Ta Quang Son 2 1 Department of Mathematics International University VNU-HCM Vietnam 2 Nhatrang Teacher College Nhatrang Vietnam Manuscript Received on May 02nd 2007 Manuscript Revised December 01st 2007 ABSTRACT Necessary and sufficient conditions for s-optimal solutions of convex infinite programming problems are established. These Kuhn-Tucker type conditions are derived based on a new version of Farkas lemma proposed recently. Conditions for s -duality and s -saddle points are also given. Keywords s -solution s -duality s -saddle point. 1. INTRODUCTION The study of approximate solutions of optimization problems has been received attentions of many authors see 6 7 9 10 11 12 and references therein . Many of these papers deal with convex problems in finite infinite dimensional spaces and finite number of convex inequality constraints and affine equality constraints. The others deal with Lipschitz problems or vector optimization problems. In order to establish approximate optimality conditions the authors often used Slater type constraint qualification see e.g. 7 11 and 12 . Recently Scovel Hush and Steinwart 13 introduced a general treatment of approximate duality theory for convex programming problems with a finite number of constraints on a locally convex Hausdorff topological vector space. In the recent years convex problems in infinite dimensional setting with possibly infinite number of constraints were studied in 2 3 where the optimality conditions duality results and saddle-point theorems were established based on the conjugate theory in convex analysis and a new closedness condition called CC instead of Slater condition. In this paper we consider a model of convex infinite programming problem that is a convex problem in infinite dimensional spaces with infinitely many inequality constraints. We study the necessary
