Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: Research Article Optimality Conditions and Duality for DC Programming in Locally Convex Spaces | Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2009 Article ID 258756 10 pages doi 2009 258756 Research Article Optimality Conditions and Duality for DC Programming in Locally Convex Spaces Xianyun Wang College of Mathematics and Computer Science Jishou University Jishou 416000 China Correspondence should be addressed to Xianyun Wang litterequation@ Received 10 February 2009 Revised 25 June 2009 Accepted 25 September 2009 Recommended by K. L. Teo Consider the DC programming problem f x - g Ax where f and g are proper convex functions defined on locally convex Hausdorff topological vector spaces X and Y respectively and A is a linear operator from X to Y .By using the properties of the epigraph of the conjugate functions the optimality conditions and strong duality of Pa are obtained. Copyright 2009 Xianyun Wang. 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. 1. Introduction Let X and Y be real locally convex Hausdorff topological vector spaces whose respective dual spaces X and Y are endowed with the weak -topologies w X X and w Y Y . Let f X R R u x g Y R be proper convex functions and let A X Y be a linear operator such that A dom f n dom g 0. We consider the primal DC difference of convex programming problem PA Xs f x - s L1 and its associated dual problem Da inf g y - f A y y eY where f and g are the Fenchel conjugates of f and g respectively and A Y X stands for the adjoint operator where Y is the subspace of Y such that y e Y if and only if A y 2 Journal of Inequalities and Applications defined by A y y A -Ỵ is continuous on X. Note that in general YA is not the whole space Y because A is not necessarily continuous. Problems of DC programming are highly important from both viewpoints of optimization theory and applications. They have been
