The Study For The Dualities Of Three Types Of The Optimization Problems

In this thesis, we study the Fenchel dualities of robust optimization prob-lem and DC composite optimization problem, and the Lagrange dualities of DC(difference of two convex functions) conical optimization problem. This the-sis consists of six chapters, the research background and the main conclusions of the robust optimization problem, DC composite optimization problem and the DC conical optimization problem arc introduced in the first chapter.In the second chapter, we provide some conceptions and lemmas.In the third chapter, we study the Fenchel dualities of the robust con-vex optimization problem. In the case when the functions are not necessarily lower semicontinuous, by using the properties of the epigraph of the conjugat-ed functions, we introduce some new constraint qualifications. Under the new constraint qualifications, we obtain characterizations of the strong duality, the stable strong duality, the total duality and the converse duality between the robust convex problem and its dual problem, which extend the results of the study of the former.In the fourth chapter, we consider the DC composite optimization problem. By using the epigraph technique and convexification technique, we give some new constraint qualifications, which completely characterize the weak duality, the zero duality and the strong duality between the prime problem and its dual problem.In the fifth chapter, we study the Lagrange dualities of the DC conical optimization problem. By using the infimal convolution of the conjugate func-tions, we present a new regularity condition for the zero duality between the problem (Pf) and its Lagrange duality.In the last chapter, we summarize the content and prospect the future research.