An iterative approach to Rank Minimization Problems

This paper investigates an iterative approach to solve the Rank Minimization Problems (RMPs) constrained in a convex set. The matrix rank function is discontinuous and nonconvex and the general RMP is classified as NP-hard. A continuous function is firstly introduced to approximately represent the matrix rank function with prescribed accuracy by selecting appropriate parameters. The RMPs are then converted to rank constrained optimization problems. An Iterative Rank Minimization (IRM) method is proposed to gradually approach the constrained rank. Convergence proof of the IRM method using the duality theory and Karush-Kuhn-Tucker conditions is provided. Two representative applications of RMP, matrix completion and output feedback stabilization problems, are presented to verify the feasibility and improved performance of the proposed IRM method.