A method for weighted projections to the positive definite cone

© 2014 Taylor & Francis. We study the numerical solution of the problem (Formula presented.) , where (Formula presented.) is a symmetric square matrix, and (Formula presented.) is a linear operator, such that (Formula presented.) is invertible. With (Formula presented.) the desired fractional duality gap, and (Formula presented.) the condition number of (Formula presented.) , we prove (Formula presented.) 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 (Formula presented.), our numerical experiments indicate excellent performance and only a very slowly growing dependence of the convergence rate on (Formula presented.). While our algorithm requires somewhat more iterations than existing interior point methods, the iterations are cheaper. This gives better computational times.