Study On Properties Of Sets And Their Application In Vector Optimization

The convexities, nonconvexities, topological properties and algebraic properties of sets have very important significance in researching on the theory and application of vector optimization. In recent years, the study of topological properties and algebraic properties of sets and their application in optimization has become one of most important subjects by means of free-disposal sets, improvement sets, co-radiant sets and assmuption B. We mainly focus on the algebraic properties of free-disposal sets and the sets which satisy assmuption B. Taking advantage of co-radiant sets, a new concept of proper efficiency is proposed and it’s Kuhn-Tucker optimality conditions are also obtained. Moreover, making use of co-radiant sets, a new definition of weak efficient solution is presented and it’s linear scalarization theorem is established. This thesis includes three chapters as follows:In chapter 1, we first give some brief introductions to the background and significance of vector optimization theory and application, especially for studying on properties of sets. And we also summarize the developments of studying on vector optimization theory and methods associated with this thesis. Secondly, we recall some basic concepts and results which will be used in this thesis. Finally, we outline the contents studied in this paper.In chapter 2, the algebraic properties of free-disposal sets are obtained. Moreover, based on the idea of Flores-Bazan and Hernandez, the topological and algebraic properties of sets are studied. We first prove that the algebraic closure is closed and the algebraic interior is open with the condition of the free-disposal sets. Meanwhile, the properties of the sum of two free-disposal sets are obtained. Secondly, int(A+B)= intA+B is proved under the assmuption B and cov(A+B)= cor A+B is also obtained under the assmuption Bi. At last, with the assumption B2, we prove that the relative algebraic interior of the sum for two sets is equal to the sum of the relative algebraic interior for these sets. With the same assumption, we also obtain that the sum of the algebraic closure of set and the relative algebraic interior of set is equal to the sum of the relative algebraic interior for the two sets. Under the assumption B3, the relative topological interior of the sum for two sets is equal to the sum of the relative topological interior for these sets, the sum of topological closure of set and the relative topological interior of set is equal to the sum of the relative topological interior for the two sets are established.In chapter 3, by means of co-radiant set we are devoted to study some unified solution concepts and some related characterizations of solutions of vector optimization problems with set-valued maps. Firstly, a kind of proper efficiency, named as C(ε)-proper efficiency, is proposed via co-radiant sets in a real locally convex Hausdorff topological linear spaces. Under the assumption of nearly C(ε)-subconvexlikeness, a Kuhn-Tucker optimality nec-essary condition is derived, by using scalarization theorem, a sufficient condition is also obtained. Moreover, under the nearly C(ε)-subconvexlikeness, an alternative theorem is established and a linear scalarization result of weakly C(ε)-efficient solutions via quasi interior is given for a class of vector optimization problems with set-valued maps.