The Stable Farkas Lemma for Composite Convex Functions in Infinite Dimensional Spaces

作     者：Gang LI Yu-ying ZHOU 

作者机构：School of Sciences Zhejiang Agriculture and Forestry University Department of Mathematics Soochow University 

出 版 物：《Acta Mathematicae Applicatae Sinica》 (应用数学学报（英文版）)

年 卷 期：2015年第31卷第3期

页      面：677-692页

核心收录：

学科分类：07[理学] 09[农学] 0903[农学-农业资源与环境] 070104[理学-应用数学] 090302[农学-植物营养学] 0701[理学-数学] 0812[工学-计算机科学与技术（可授工学、理学学位）] 

基　　金：Supported by the Natural Science Foundation of China(Nos.11401533,11301484,11071180,11371273,11171247) Hong Kong Polytechnic University(G-YX1Q) the Research Grants Council of Hong Kong(No.Poly U 5306/11E) Scientific Research Foundation of Zhejiang Agriculture and Forestry University(No.2013FR080) 

主　　题：conjugate functions epigraph stable Farkas lemma stable duality 

摘      要：In this paper, we consider a general composite convex optimization problem with a cone-convex system in locally convex Hausdorff topological vector spaces. Some Fenchel conjugate transforms for the composite convex functions are derived to obtain the equivalent condition of the Stable Farkas Lemma, which is formulated by using the epigraph of the conjugates for the convex functions involved and turns out to be weaker than the classic Slater condition. Moreover, we get some necessary and sufficient conditions for stable duality results of the composite convex functions and present an example to illustrate that the monotonic increasing property of the outer convex function in the objective function is essential. Our main results in this paper develop some recently results.