Abstract Convexity Of Vector And Set-valued Topical Functions With Applications

In this thesis,a typical abstract convex function,the so called topical function is extended and investigated in the frame of abstract convexity.First,the efficient solutions and weakly efficient solutions for a vector optimization defined by some vector topical function are studied by the approach of scalarization.After generalizing the support set to the vector case,we introduce a new type of vector topical function as well as a set-valued one,and establish the abstract convex theories for them.Based on this,we consider some conjugate duality for the DC type constrained optimization defined by the topical functions.Then,we propose a Lagrange type conjugate duality for the constrained extreme optimization,and then study the dual theory by the aid of the image space analysis.Also,using the concept of set-valued topical function given above,we generalize this duality to the vector situation.The thesis contains six chapters and it is organized as follows:In Chapter 1,the development and researches on the theories of optimization problems are briefly introduced.Then researches about abstract convexity,topical function,scalarization,image space analysis and conjugate dual theory are reviewed.In the end,the motivations and the main work in this thesis are given.In Chapter 2,some notations,assumptions,basic definitions and properties involved in this thesis are presented.Mainly refer to the abstract convex analysis,topical function,the nonlinear scalarization function proposed by Gerstewitz(Tammer),the supremum concept introduced by Tanino and the separation functions for image space analysis.In Chapter 3,a vector constrained optimization involving some kind of vector topical function is investigated by the approach of scalarization.First,we consider some scalarization functions,which consist of the Gerstewitz function,the symmetric form of it and their convex combinations.We investigate the abstract convexity properties of these scalarization functions and use them to identify the maximal points of a set in an ordered vector space.Then,we exploit them to scalarize the topical vector optimization and transform it into some equivalent inequality system.We establish some versions of Farkas type results for the inequality system by means of the epigraph of conjugate function,the support set and the polarity theory.Finally,we use these results to characterize the efficient solutions and weakly efficient solutions for the vector topical optimization problem.In Chapter 4,the extensions for the concept and theories of topical function to the vector and set-valued cases are considered.First,we introduce a new type of vector topical function.It contains some other categories of topical functions as special cases and can be interpreted as weak separation functions in image space analysis.We establish its envelope result and investigate its properties in the frame of abstract convexity.Then,we present the corresponding conjugation and subdifferential,and observe the relationships among these concepts.Furthermore,based on the ideas above,we introduce a concept of set-valued topical function,and studied its abstract convex theory as well.In Chapter 5,making use of the theories of topical functions,we study the conjugate dualities for some classes of constrained optimizations.First,we obtain some dual results for a vector optimization,where the object is expressed as the difference of vector topical functions.Also,the set-valued case is considered.Then,the Lagrange type conjugate duality for the constrained optimizations is investigated in the frame of abstract convexity.With a perturbation in the constraints,we introduce a Lagrange type duality.Inspired by some ideas of the image space analysis,we introduce the corresponding separation functions to study the duality for the general constrained extreme optimization.Some equivalent characterizations and sufficient conditions to zero duality gap as well as the strong duality are obtained.Then,we pay special attentions to the constrained optimizations with some topical features.It turns out that this class of optimizations reaches the zero duality gap easily.With the aid of the extensions for the notions and theories of topical function to the set-valued case in Chapter 4,we also consider such a duality for the vector constrained optimizations.The zero duality gap and the strong duality for this vector duality are investigated.In Chapter 6,we briefly summarize the main results of this thesis,and propose some problems for further investigation.