Generalized Convexity In Convex Metric Space

Convex sets and convex mappings,generalized convex sets and generalized convex mappings are one of the main research contents of convexity theory and generalized convexity theory respectively.They have very important applications in inequality theory,linear and nonlinear programming,constrained and unconstrained optimization,multi-objective programming,game theory,fixed point theory,nonlinear analysis and so on.W-convexity sets and W-convexity functions in convex metric spaces are some generalizations of standard convexity sets and standard convexity functions in linear spaces,they are two types important generalized convexity.The purpose of this paper is to study some W-convexity sets and W-convexity functions and their applications in convex metric spaces.In chapter 1,we briefly introduce the research status of generalized convexity in linear spaces and generalized convexity in convex metric spaces.In chapter 2,some properties of W-convex sets in convex metric spaces are discussed.Two concepts of generalized convex sets based on standard convex structure in linear spaces are introduced into convex metric spaces,and two new generalized W-convex sets(nearly W-convex sets and weakly nearly W-convex sets)are defined.Some properties of new generalized W-convex sets are studied.In chapter 3,three concepts of generalized convex functions based on standard convex structure in linear spaces are introduced into convex metric spaces,and three new generalized W-convex functions(W-semi-strictly convex function,W-strictly quasiconvex function and W-semi-strictly quasiconvex function)are defined.Some relationships between W-convexity functions and the epigraphs or the level sets are studied.Secondly,under appropriate conditions,it is proved that the intermediate point W-convex function is a midpoint W-convex function,and the midpoint W-convex function is [0,1](?)Q-W-convex function.Finally,a density theorem is obtained,and some applications of the density theorem in minimization and multi-objective programming problems are discussed.In chapter 4,using the generalized W-convex function newly defined in chapter3,some criterions for W-convex functions are established under the conditions of midpoint W-convexity and upper semi-continuity or lower semi-continuity or W-quasiconvexity or W-strictly quasiconvexity or W-semi-strictly quasiconvexity.In chapter 5,we summarize the full text and the future research direction,and propose two conjectures.