Optimality Conditions And Duality Theory For Two Types Of Semidefinite Programming

Semidefinite programming is an important branch of mathematical programming. In recent years, it has gained rapid development both in theory and algorithm. Thus various forms of semidefinite programming problems have emerged. Nonconvex semidefinite pro-gramming is one of the important forms of semidefinite programming. It is widely applied to perturbation analysis, control theory, electronic engineering and other fields. Multi-objective semidefinite programming is a kind of mathematical programming which study includes semidefinite constraint matrix function and multiple objective function. It is an effective combination of multiobjective programming and semidefinite programming. So nonconvex semidefinite programming and multiobjective semidefinite programming have important research significance and application value.It is well known, in the optimization problem, the optimality conditions and duality theory are very important research subjects. This paper focuses on the optimality condi-tions and duality theory of the nonconvex semidefinite programming and multiobjective semidefinite programming. To specific:1. In chapter 2, It is concentrated on nonconvex semidefinite programming. Firstly, a sufficient condition for nonconvex semidefinite programming is presented under the invex-ity assumption. I discussed a second order sufficient condition of nonconvex semidefinite programming without any convexity assumption. Secondly, a sufficient and necessary condition for the KKT condition is given, and the necessary condition of nonconvex semidefinite programming is obtained by applying the result to the problem. Finally, I study the optimality conditions of saddle point. The saddle point type sufficient and necessary condition and sufficient condition are given.2. In chapter 3, I focus on multiobjective semidefinite programming. First, the nec-essary condition for multiobjective semidefinite programming is obtained by using the necessary condition of nonconvex semidefinite programming. Furthermore, under invex-ity assumption, the optimality sufficient condition is presented. Finally, the Wolfe dual, Lagrange dual programming under the means of the efficient s(?)tion and Lagrange dual under the means of the weak efficient solution are established. The corresponding duality theorems are derived, including weak duality, strong duality, inverse duality and saddle point optimality conditions.