Research On Extra-gradient Method For Semidefinite Programming

Compared with linear programming, vector is replaced by matrix and non-negativity of vector is replaced by semidefinite property of matrix in semidefinite programming.Thus,Semidefinite programming is an extension of linear programming. There are many methods for solving semidefinite programming, in which the interior-point methods are the most successful. Most interior-point methods for linear programming have been generalized to linear semidefinite programming. Semidefinite programming is a branch of convex optimization problem. Important applications of semidefinite programming exist in combinatorial optimization, approximation theory, system and control theory, and mechanical and electrical engineering. Investigating semidefinite programming is very significative both in theory and in practice.In this paper , semidefinite programming is effectively transformed into a variational inequality problem. An improved extra-gradient method is proposed, then semidefinite programming is sloved.For detail, we conclude them as follows:The relations of some important optimization problems are systematically studied in this paper, such as linear programming, convex quadratic programming, second-order cone programming, semidefinite programming and polynomial optimization problems. When semidefinite programming satisfies the Slater conditions, linear semidefinite programming and Quadratic Semidefinite Programming are transformed into variational inequality problems. In this paper, a new strategy for computing correction step size is proposed in general extra-gradient method. Further, the convergence of the new algorithm is proved. Numerical experiment shows that the new method is effective for solving semidefinite programming.