The Lagrange Dualities Of Composite Optimization Problem

In this paper, we study the Lagrange dualities of convex composite opti-mization problem and DC composite optimization problem. This paper consists five 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 study the Lagrange dualities of convex composite optimization problem. In the case when the involved functions are not neces-sarily lower semicontinuous and the sets are not necessarily closed, we introduce some new constraint qualifications by using the properties of the epigraph of the conjugated functions and the properties of subdifferentials. Under the new constraint qualifications, the strong duality, the stable strong duality, the to-tal duality and the stable total duality between convex composite optimization problem and its dual problem are established. Our results extend and improve the corresponding results in the previous papers.In the fourth chapter, we consider the Lagrange dualities of DC composite optimization problem. In the case when the functions are not necessarily lower semicontinuous and the sets are not necessarily closed, by using the epigraph and convexification technique, we give some new constraint qualifications. Under the new constraint qualifications, some necessary and sufficient conditions ensuring of the weak duality, the zero duality and the strong duality to hold between the problem DC composite optimization problem and its dual problem are given.In the last chapter, we summarize the content and prospect the future research.