Two Relaxation Methods for Rank Minimization Problems

The problem of minimizing the rank of a symmetric positive semidefinite matrix subject to constraints can be lifted to give an equivalent semidefinite program with complementarity constraints. The formulation requires two positive semidefinite matrices to be complementary. This is a continuous and nonconvex reformulation of the rank minimization problem. We develop two relaxations and show that constraint qualification holds at any stationary point of either relaxation of the rank minimization problem, and we explore the structure of the local minimizers.