A New Method For Solving The Linear Semidefinite Programming

In recent years, semidefinite programming (SDP), a very active research topic, is arisingin the operations research. It makes many new challenges for specialists in optimization whilepromoting greatly the developments and applications of optimization. Semidefiniteprogramming has wide applications in combinatorial optimization, control theory,eigenvalue optimization, optimal design, electrical engineering and statistics.Interior-point method is the most successful method for solving the semidefiniteprogramming. Primal-dual interior-point method is a kind of the interior-point method, and isused widely for its effectivity no matter in practice or theory. But some problems still exist,for example it requires the initial point is feasible. In this paper, a homotopy method forsolving the linear semidefinite programming is given. Firstly, it constructs a homotopyequation via the KKT condition of linear semidefinite programming, then the existence andconvergence of smooth homotopy path are proved. The initial point can be chosen to be anyinterior point and need not to be a feasible point.