Optimality Conditions For Composite Optimization Problems With Conical Constraints

In this paper,we study the optimality conditions of convex composite optimization problem and DC composite optimization problem.This paper consists four chapters.In the first chapter,the research background and the main conclusions for convex composite optimization problem and for DC composite optimization problem are introduced.In the second chapter,we provide some notations,conceptions and lemmas.In the third chapter,we mainly study optimality conditions,saddle point theorems and the Fenchel-Lagrange dualities of convex composite optimization problem.In the case when the involved functions are not necessarily lower semi-continuous and the sets are not necessarily closed,we introduce some new con-straint qualifications by using the properties of the epigraph of the conjugated functions and the properties of subdifferentials,which characterize completely the optimality conditions for composite optimization problem with conical con-straints.Moreover,saddle point theorems for the Lagrange function of convex composite optimization problem are also obtained.Under the new constraint qualifications,the strong duality,the stable strong duality,the zero duality,the stable zero duality and the stable total duality between convex composite opti-mization problem and its dual problem are also established.Our results extend the corresponding results in the previous papers.In the fourth chapter,we consider the optimality conditions and saddle point theorems of two kinds of DC composite optimization problems.In the case when the functions are not necessarily lower semicontinuous and the sets are not necessarily closed,some new constraint qualifications are obtained by using the epigraph and convexification technique.Under those constraint qual-ifications,some optimality conditions for two kinds of DC composite optimiza-tion problems with conical constraints are established.Moreover,saddle point theorems for the Lagrange function of the two kinds of optimization problems are also obtained.