(Directional) Derivatives Of Three Classes Of Set-Valued Maps And Their Applications To Optimization

This dissertation studies mainly approximations to special classes of set-valued maps and their applications, in order to compute differentials of some class of set-valued maps and to solve basis theories of constructing high-ordered methods of nonsmooth functions. Then results obtained in this dissertation are applied to optimality theories in optimization. The main results obtained in this dissertation are summarized as follows:1. Chapter 2 established derivatives of a class of set-valued maps and differentials of subdifferential maps of convex functions in the sense of Tyurin (1965) and Banks & Jacobs (1970) based on theories of convex pairs space.2. Chapter 3 applied the results obtained in Chapter 2 to estimate to solution-set of a parametric mathematical programming. Results about stability of a parametric linear programming and estimate to bound of solution-set of a parametric linear programming is established; under the assumptions in this Chapter, results obtained is more sharper than ones obtained in last.3. Chapter 4 is devoted to the study of diferential structure in Quasidiferential analysis-quasidiferential structure. This chapter proposes three conceptions, i.e., Kernelled quasidiferential, star-kernel and star-diferential, and establishes their operational properties. A sufficient theorem and a sufficent and necessity theorem for a quasi-kernel being a kernelled quasidiferential are proven. Both the existence of star-kernel for a quasidiferentiable function and the existence of star-differential for a direnction-ally diferentiable function are established. The relationships between sub- and super-derivatives and Penot diferentials are dicussed as well.4. In recent years, set-valued optimization make much progress. In Chapter 5, based on Clarke tangent cone, we establish epiderivative of a class of set-valued maps and its properties. And furthermore, sufficiency (or neccessity) optimization conditions of set-valued optimization are also obtained.