A New Projection Contraction Algorithm for Semidefinite Programming

Abstract

This paper studies a projective contraction algorithm for solving semidefinite programming problems. The interior point algorithm is a classic algorithm for solving semidefinite programming problems. It can be divided into feasible and infeasible algorithms according to whether the initial point is feasible or not. The feasible interior point algorithm needs to use a self-dual strategy to find strictly feasible points. Although the infeasible interior point algorithm does not require the initial point to be strictly feasible, the theoretical analysis of the algorithm is very complicated. Compared with the interior point algorithm, the projection contraction algorithm has no requirement on the initial point, and the algorithm analysis is much simpler. The optimal condition of the semidefinite programming problem can be rewritten as a projection equation, which is then solved by the projection contraction algorithm. We propose a new projection contraction algorithm based on different descent directions to solve the semidefinite programming problem. Further, under the framework of the projection contraction algorithm, an improved algorithm is proposed, that is, changing the descending direction by solving the linear equation system, thus avoiding the high-dimensional inversion operation so that the algorithm can solve the large-scale semidefinite programming problem. Numerical experiments show that the algorithm is feasible and the improved algorithm is competitive with the interior point algorithm.