Generalized Differentiability And Lipschitz Properties Of Solutions For Vector Optimization Problems

In this thesis, the Aubin property and contingent derivatives for a class of implicit multifunctions are studied. And the relationships between the implicit multifunctions and the solution mappings of kinds of vector optimization problems are established. Then the Aubin property and contingent derivatives for the solution mappings of kinds of vector optimization problems are obtained. This thesis is divided into eight chapters. It is organized as follows:In Chapter 1, the development and current researches on the topic of Lipschitz properties, H?lder continuity and sensitivity for vector optimization problems are firstly recalled. Then, the development and current researches for generalized equations and implicit multifunctions are reviewed. Finally, the motivations and the main research work are also given.In Chapter 2, some basic notions and definitions of vector optimization problems are recalled. These definitions mainly refer to upper (lower) semicontinuity, Lipschitz properties, metric regularity, Robinson's metric regularity and kinds of graph deriva-tives.In Chapter 3, a computing rule of the second order contingent derivative for a set-valued mapping and a convex cone is firstly obtained. Then, the second order con-tingent derivative of a set-valued gap function for a weak vector variational inequality is discussed.In Chapter 4, a set-valued gap function of a parametric weak vector equilibrium problem which is an optimal value mapping is firstly introduced. And the explicit ex-pression of contingent derivative for the optimal value mapping is established. Then, the explicit expression of contingent derivative for an implicit multifunction is obtained. Finally, these results are applied to the solution mappings of kinds of parametric vector optimization problems.In Chapter 5, by using the contingent derivative of a field mapping, the Robinson's metric regularity and Aubin property of an implicit multifunction are obtained. Then, make using of Ekeland variational principle, the Robinson's metric regularity and Au-bin property of another implicit multifunction are discussed. These results are very use-ful for investigating the Robinson's metric regularity and Aubin property for kinds of parametric vector optimization problems. In Chapter 6, by decomposing a parametric weak vector variational inequality, the solution mapping of the problem can be written as an implicit multifunction. By apply-ing the results of Chapter 5, the Robinson's metric regularity and Lipschitzian stability of the solution mapping for the parametric weak vector variational inequality are firstly established. Then the contingent derivative of the solution mapping for the parametric weak vector variational inequality is discussed. Finally, the Robinson's metric regularity, Aubin property and the contingent derivative of the solution mapping for a parametric vector optimization problem are also considered.In Chapter 7, since the set-valued gap function of the parametric weak vector equi-librium problem is complex, it is difficult to consider the Robinson's metric regularity and Aubin property of the solution mapping for the problem. In this chapter, a scalar gap function is utilized to discuss the Robinson's metric regularity and H?lder-likeness of the solution mapping for the problem. Then the contingent derivative of the solution mapping is also obtained which is different from that of Chapter 4.In Chapter 8, the results of this thesis are briefly summarized. And some problems which are remained and thought over in future are put forward.