An Inexact ABCD Method Based Majorized Penalty Approach For Rank Constrained Quadratic Semidefinite Programming Problems

In this paper, we aim at solving the rank constrained quadratic semidefinite programming (rank-QSDP) problem. The problem is a non-convex problem due to the existence of the rank constraint. We first penalize the rank constraint to the objective function, and majorize the problem to a least squares one. And then we may solve the problem sequentially. In order to solve the reformulated problem to desired accuracy efficiently, we adopt an inexact accelerated block coordinate descent (ABCD) method to solve the inner problem. Furthermore, at the beginning we solve a simplified version of the problem to generate a good initial point, and during this process we penalize the nuclear norm of the variable instead of the majorized function of the rank constraint. Numerical experiments demonstrate that method is very efficient, especially for rank-QSDP problems with many constraints.