Nonlinear Scalarization Functions And Their Applications In Vector Optimization Theory

A vector optimization problem refers to minimization of a vector-valued function under some constrained conditions. During the rapid development of vector optimization theory in the last decades, it has been associated tightly with many branches in mathe-matics and economics. So far, vector optimization theory has already turned into a huge system, which not only involves many different aspects and directions but also contains plentiful contents and results.Since the scalar optimization theory and methods have made great progress until now, scalarization which solves vector optimization problems by converting them into scalar ones has been proved an important and effective approach. Linear scalarization is easy and simple, however convexity or generalized convexity of objective function on feasible set is absolutely necessary, which to some extent has limited its applications. Consequently, in order to deal with more nonconvex vector optimization problems in practice, many researchers have gradually concentrated on the study of nonlinear scalarization in vector optimization theory. Thereinto, the key task is that how to find or construct a suitable nonlinear scalarization function.Our work is concerned with properties and applications of nonlinear scalarization functions in vector optimization theory, and the main contributions could be divided into six parts as follows:Firstly, we investigate some fundamental properties of a nonlinear scalarization func-tion named by the biggest strictly monotonic function and also give its dual form cor-responding to Gerstewitz functionals. Then the concepts of cone-shaped neighborhoods and a new class of cone-semicontinuity are introduced in the following. Besides, by us-ing two nonlinear scalarization functions including Gerstewitz functional and the biggest strictly monotonic function, we have obtained complete and unified characterizations of cone-semicontinuity for vector-valued maps.Secondly, by these two nonlinear scalarization functions above, we construct a semi- norm and a related normed linear space under a kind of equivalent relation. In addition, based on usual strict efficiency and superefficiency, we introduce new notions of cone-strict efficiency and cone-superefficiency. The relationships among them are established. Characterizations of cone-strict efficiency are also obtained, which refer to well-posedness of the corresponding scalarization problem.Thirdly, the classical notions of augmented dual cones in normed linear spaces are extended to locally convex spaces. The extended augmented dual cones are defined sep-arately in two cases by the semi-norms. Then we discuss their main properties, and also establish existence conditions of nontriviality of the extended augmented dual cones under suitable hypothesis. Moreover, the concepts of extended augmented dual cones respec-tively with respect to Gerstewitz functional and the biggest strictly monotonic function are proposed in a Hausdorff topological vector space. We also present some properties and existence theorems for the nontriviality of them.Fourthly, by the notions of base functionals and augmented dual cones, we indi-cate firstly that the norms, Gerstewitz functionals and oriented distance functions have common characteristics with base functionals. After that, the equivalence among these three sublinear functions on the ordering cone is established by using the structures of augmented dual cones under the assumption that the ordering cone has a bounded base. However we show that two superlinear functions do not have similar relations with the norm ahead. More generally, the equivalence among three sublinear functions outside the negative cone has also been obtained in the end.Fifthly, characterizations of proper cone-quasiconvexity for vector-valued maps are provided separately by a strict lower level set and the biggest strictly monotonic function. Then we focus on the discussion of proper cone-quasiconvexity of set-valued maps in this work. Based on two ordinary set-order relations in a real topological vector space, characterizations of properly quasiconvex set-valued maps are established respectively. The approaches we use here contain two different kinds of level sets and the biggest strictly monotonic function.Finally, we introduce the notion of scalar cone-quasiconvexity for set-valued maps based on a kind of set-order relation appearing in the last chapter. Also the relationships among several cone-convexities are discussed. Meanwhile, the scalarization composition rule of cone-convexity of set-valued maps is established by a real-valued increasing convex function. We obtain a characterization for cone-quasiconvexity of set-valued maps by using Gerstewitz functional in the end.