Inexact primal-dual path-following algorithms for a special class of convex quadratic SDP and related problems

We propose a primal-dual path-following Mehrotra-type predictor-corrector method for solving convex quadratic semidefinite programming (QSDP) problems. For the special case when the quadratic term has the form 1 2 X •(UXU), we compute the search direction at each iteration from the Schur complement equation. We are able to solve the Schur complement equation efficiently via the preconditioned symmetric quasiminimal residual iterative solver with two appropriately constructed preconditioners. Numerical experiments on a variety of QSDPs with matrices of dimensions up to 2000 are performed and the computational results show that our methods are efficient and robust. Our methods can also be extended to linear SDP problems with upper bound constraints on primal matrix variables.