Two Numerical Methods. Solutions Semidefinite Programming

Since semi-definite programming can be widely applied in many fields, such as combinatorial optimization, electrical engineering, so the research on semi-definite programming has become a very active research in recent years. In recent years, the theory and algorithms for semi-definite programming have made great progress.The numerical methods of semi-definite programming were studied in this paper. There sections were included in this paper.In the first section, the basic theory and conclusion of semi-definite programming were introduced, such as its dual theory and optimal conditions.In the second section, a new improved kernel function for the general SDP problems was proposed. Its related properties were studied. A primal-dual interior point algorithm for semi-definite programming based on the kernel function was presented. The iteration bounds corresponding to long-step and short-step methods were obtained.In the third section, a new nonlinear Lagrange function for the nonconvex semi-definite programming problem was proposed. Its related algorithm and properties were studied. And the convergence of the algorithm was proved. It was proved that the sequence generated by Lagrangian was locally convergent when the penalty parameter was larger than a threshold under suitable conditions. Thereby the error estimates of the solution was obtained which depended on the penalty parameter. The feasibility and effectiveness of the algorithm were also studied.