Duality Theories And Optimality Conditions For Robust Composite Optimization Problems

In this paper,under the assumptions that the set is not necessarily closed and the function is not necessarily lower semicontinuous,we study the duality theories and optimality conditions of robust composite optimization problem.This paper is divided into four chapters.In the first chapter,the research background and the main conclusions for robust composite optimization problem are introduced.In the second chapter,we provide some notations,conceptions and lemmas.In the third chapter,we study the zero duality and the strong duality of the robust composite optimization problem.By using infimal convolution property and the epigraph technique of the conjugate functions,we give some new constraint qualifications.Under these constraint qualifications,the zero duality,the strong duality,the stable zero duality and the stable strong duality between robust composite optimization problem and its Lagrange dual problems are established,which extend the corresponding results in the previous papers.In the fourth chapter,we study the total duality and the optimality condi-tion of the robust composite optimization problem.By using the subdifferential property of the function,two new constraint qualifications are introduced.Un-der these constraint qualifications,the total duality and the stable total duality between robust composite optimization problem and its Lagrange dual problems are established,and the equivalent characterization of the optimality condition of the solution are provied.