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 for Approximate Solutions in Multiobjective Optimization Problems | Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2010 Article ID 620928 17 pages doi 2010 620928 Research Article Optimality Conditions for Approximate Solutions in Multiobjective Optimization Problems Ying Gao 1 Xinmin Yang 1 and Heung Wing Joseph Lee2 1 Department of Mathematics Chongqing Normal University Chongqing 400047 China 2 Department of Applied Mathematics The Hong Kong Polytechnic University Hung Hom Kowloon Hong Kong Correspondence should be addressed to Ying Gao gaoyingimu@ Received 18 July 2010 Accepted 25 October 2010 Academic Editor Mohamed El-Gebeily Copyright 2010 Ying Gao et al. 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. We study first- and second-order necessary and sufficient optimality conditions for approximate weakly properly efficient solutions of multiobjective optimization problems. Here tangent cone e-normal cone cones of feasible directions second-order tangent set asymptotic second-order cone and Hadamard upper lower directional derivatives are used in the characterizations. The results are first presented in convex cases and then generalized to nonconvex cases by employing local concepts. 1. Introduction The investigation of the optimality conditions is one of the most attractive topics of optimization theory. For vector optimization the optimality solutions can be characterized with the help of different geometrical concepts. Miettinen and Makela 1 and Huang and Liu 2 derived several optimality conditions for efficient weakly efficient and properly efficient solutions of vector optimization problems with the help of several kinds of cones. Engau and Wiecek 3 derived the cone characterizations for approximate solutions of vector optimization problems by using translated cones. In 4 Aghezzaf and Hachimi obtained second-order .
