The Non-interior Algorithm For Semidefinite Programming

Semidefinite programming is an extension of linear programming. In semidefinite programming, one maximizes (minimizes) a linear function subject to the constraint that an affine combination of symmetric matrices is positive semidefinite. Such a constraint is nonlinear and nonsmooth, but convex, so semidefinite programming is a nonsmooth and convex optimization problem.In the paper, the theory, algorithm,and recent reseach of semidefinite programming are summarized, some works in algorihm are introduced. In detail, we conclude them as follows:1. A filter method for semidefinite programming is proposed in this paper. Firstly, a low-rank decomposition technique is adopted to transform the standard semidefinite programming into an equivalent nonlinear programming problem. Then two kinds of filter algorithms are introduced to solve the problem. The first algorithm is based on a decomposition of direction. We give the main idea of it and the convergence conclusions. The second is a combination of filter and the SQP. A detailed analysis is carried out and better global convergence conclusions are obtained.2. An extensive Fischer-Burmeister function method are used to convert the optimality conditions of SDP into an equivalent nonsmooth system of equations, then by smoothing the nonsmooth equations, a smoothing-type method for semidefinite programming is constructed. At last, on the basis of the superlinear convergence, an improved algorithm is presented and a rapid quadratic convergence conclusion is obtained.