Semidefinite Programming Problem Two Numerical Solution

Optimality conditions and numerical methods of nonlinear semidefinite programming were studied in this paper which included the following three parts.The first chapter described the basic knowledge of semidefinite programming, outlined optimality conditions, which were prepared for the Lagrange method in the next chapter, and on this basis, we derived optimality conditions for a class of non-smooth semidefinite programming problem with equal constraints.In the second chapter, a nonlinear Lagrange function for general nonconvex semidefinite programming was given. We analyzed the characteristic of this Lagrange function on the KKT point. Under some conditions, the local convergence of the method based on this function was proved that there exists a threshold of the penalty parameter such that the sequence products of the algorithm locally converge to the KKT point when the penalty parameter is more than the threshold. Besides, the error bound of the solutions with the penalty parameter was estimated. Numerical examples also demonstrated the execution and effectiveness.In the third chapter, an analytic center cutting plane method was given for solving the large-scale semidefinite programming. The convergence of this method was proved, and eventually the practical example was reported.