TAILIEUCHUNG - Báo cáo hóa học: "ON D-PREINVEX-TYPE FUNCTIONS"

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: ON D-PREINVEX-TYPE FUNCTIONS | ON D-PREINVEX-TYPE FUNCTIONS JIAN-WEN PENG AND DAO-LI ZHU Received 7 April 2006 Revised 10 July 2006 Accepted 26 July 2006 Some properties of D-preinvexity for vector-valued functions are given and interrelations among D-preinvexity D-semistrict preinvexity and D-strict preinvexity for vector-valued functions are discussed. Copyright 2006 . Peng and . Zhu. 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 and preliminaries Convexity and generalized convexity play a central role in mathematical economics engineering and optimization theory. Therefore the research on convexity and generalized convexity is one of the most important aspects in mathematical programming see 1-4 6-11 and the references therein . Weir and Mond 7 and Weir and Jeyakumar 6 introduced the definition of preinvexity for the scalar function f X c R R. Recently Yang and Li 9 gave some properties of preinvex function under Condition C. Yang and Li 9 introduced the definitions of strict preinvexity and semistrict preinvexity for the scalar function f X c R R and discussed the relationships among preinvex-ity strictly preinvexity and semistrictly preinvexity for the scalar functions. Yang 8 also obtained some properties of semistrictly convex function and discussed the interrelations among convex function semistrictly convex function and strictly convex function. Throughout this paper we will use the following assumptions. Let X be a real topological vector space and Y a real locally convex vector space let s c X be a nonempty subset let D c Y be a nonempty pointed closed convex cone Y is the dual space of Y equipped with the weak topology. The dual cone D of cone D is defined by D f e Y f y f y 0 Vy e D . From the bipolar theorem we have the following. Lemma . For all q e D q d 0 if and only if d e D. Hindawi .

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