Stability Of Solutions For Vector Optimization Problems In Infinite-Dimensional Spaces

In recent years, as an important research branch of optimization theory and applications, vector optimization theory and methods had been developed rapidly and became one of main research fields in optimization. The study on which involves many disciplines, such as convex analysis, non-smooth analysis, non-linear analysis and so on. At the same time, it has wide applications in economics, management, engineering design, transportation, environmental protection and optimal design, etc. In this paper, we are going to study stability of solutions for vector optimization problems in infinite-dimensional space. We obtain some stability results for vector optimization problems via variational analysis method. It is organized as follows:In Chapter 1, we first introduce the background and significance of vector optimization theory and its applications, including the development of vector optimization problems. Then, we give an outline of present situation about variational analysis methods, including its applications to study of existence, nonemptiness and boundedness, stability for vector optimization problems. Finally, we list some basic conceptions and lemmas which are used in this dissertation.In Chapter 2, we study the variational convergence properties of vector-valued functions in the infinite dimensional space by means of variational analysis method. First of all, we define the notion that (Ck,fk)â†'(C,f) in the infinite-dimensional space, which extend the notion of variational convergence in finite-dimensional space already. Second, we characterize the variational convergence via the set convergence of epigraphs and coepigraphs when (Ck, fk) â†' (C, f) in the infinite-dimensional space. Then we give several equivalent characterication notions of (Ck, fk) â†' (C,f), which extend some conclusion in the finite-dimensional space. In the end, we use the notion and properties of the level sets to get some other equivalent characterications of (Ck, fk) â†' (C, f).In Chapter 3, we study the stability of the vector optimization problem VOP(C, f) and VOP(Ck, fk) by using established conclusion in Chapter 2. Where VOP(C, f) is to find x ∈ C such that f(y)-f(x) (?)-int P, (?)y ∈ C. And VOP(Ck, fk) is to find xk ∈C such that fk(y)-fk(xk)(?)-int P,(?)y ∈ Ck. We first define the notion of asymptotic function to vector-valued functions in the finite-dimensional space, which extend the notion of asymptotic function for scalar-functions in finite-dimensional space. Then we employ the asymptotic function and asymptotic cone to define two important sets Qω and Rω, and study the stability of Qω, Rω, and level sets for vector-valued functions. We establish the variational convergence properties for Qω, Rω, and level sets. Finally, we obtain the stability of solutions for vector optimization problems by using the properties of variational convergence for Qω, Rω, and level sets in infinite-dimensional space. As an application, we study the stability properties of efficient and weakly efficient minimal points for set sequences Ak when Akâ†' A, and obtain variational convergence results for its solutions.