tailieunhanh - Báo cáo hóa học: " Research Article Tightly Proper Efficiency in Vector Optimization with Nearly Cone-Subconvexlike "

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 Tightly Proper Efficiency in Vector Optimization with Nearly Cone-Subconvexlike | Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2011 Article ID 839679 24 pages doi 2011 839679 Research Article Tightly Proper Efficiency in Vector Optimization with Nearly Cone-Subconvexlike Set-Valued Maps Y. D. Xu and S. J. Li College of Mathematics and Statistics Chongqing University Chongqing 401331 China Correspondence should be addressed to Y. D. Xu xyd04010241@ Received 26 September 2010 Revised 17 December 2010 Accepted 7 January 2011 Academic Editor Kok Teo Copyright 2011 Y. D. Xu and S. J. Li. 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. A scalarization theorem and two Lagrange multiplier theorems are established for tightly proper efficiency in vector optimization involving nearly cone-subconvexlike set-valued maps. A dual is proposed and some duality results are obtained in terms of tightly properly efficient solutions. A new type of saddle point which is called tightly proper saddle point of an appropriate set-valued Lagrange map is introduced and is used to characterize tightly proper efficiency. 1. Introduction One important problem in vector optimization is to find efficient points of a set. As observed by Kuhn Tucker and later by Geoffrion some efficient points exhibit certain abnormal properties. To eliminate such abnormal efficient points there are many papers to introduce various concepts of proper efficiency see 1-8 . Particularly Zaffaroni 9 introduced the concept of tightly proper efficiency and used a special scalar function to characterize the tightly proper efficiency and obtained some properties of tightly proper efficiency. Zheng 10 extended the concept of superefficiency from normed spaces to locally convex topological vector spaces. Guerraggio et al. 11 and Liu and Song 12 made a survey on a number of definitions of proper .