A method for weighted projections to the positive definite cone

Abstract

We study the numerical solution of the problem , where is a symmetric square matrix, and is a linear operator, such that is invertible. With the desired fractional duality gap, and the condition number of , we prove iteration complexity for a simple primal-dual interior point method directly based on those for linear programs with semi-definite constraints. We do not, however, require the numerically expensive scalings inherent in these methods to force fast convergence. For low-dimensional problems (), our numerical experiments indicate excellent performance and only a very slowly growing dependence of the convergence rate on . While our algorithm requires somewhat more iterations than existing interior point methods, the iterations are cheaper. This gives better computational times.