Optimality Conditions And Value Function Estimations For Composite Optimization Problems Based On Generalized Subdifferentials

In this paper,the optimality conditions and value functions of composite optimization problems with infinite inequality constraints are studied.This paper is divided into five chapters.The first chapter mainly introduces the research background of convex composite optimization problem and DC composite optimization problem and some main conclusions of this paper.The second chapter gives some notations and conceptions.In chapter 3,we study the optimality conditions for the composite optimization of DC with one variable.In the case when the involved functions are not necessarily continuous and the sets are not necessarily closed,we introduce some new constraint qualifications by using the properties of the epigraph and subdifferentials of the functions and convexification technique.Under this new constraint qualifications,the local and global sufficient and necessary optimality conditions for this optimization problem are established.In chapter 4,we study the optimality conditions for the convex composite optimization and composite optimization of DC with binary variable.Characterizations of optimal solutions of two kinds of composite optimization problems are described based on advanced tools of variational analysis,which extend the corresponding results in the previous papers.In chapter 5,we study the subdifferentials of value functions for binary DC composite optimization problems.In the case when the involved functions are not necessarily continuous and the sets are not necessarily closed,some constraint qualifications are obtained by using the properties of the generalized subdifferentials of the functions.We establish the upper estimators of Frechet subderivatives and Mordukhovich subdifferentials of value functions.