A Customized ADMM for Rank-Constrained Optimization Problems with Approximate Formulations

This paper proposes a customized Alternating Direction Method of Multipliers (ADMM) algorithm to solve the Rank-Constrained Optimization Problems (RCOPs) with approximate formulations. Here RCOP refers to an optimization problem whose objective and constraints are convex except a (nonconvex) matrix rank constraint. We first present an approximate formulation for the RCOP with high accuracy by selecting an appropriate parameter set. Then a general ADMM frame is employed to solve the approximated problem without requiring singular value decomposition in each subproblem. The new formulation and the customized ADMM algorithm greatly enhance the computational efficiency and scalability. While ADMM has been extensively investigated for convex optimization problems, its convergence property is still open for nonconvex problems. Another contribution of this paper is to prove that the proposed ADMM globally converges to a stationary point of the approximate problem of RCOP. Simulation examples are provided to demonstrate the feasibility and efficiency of the proposed method.