A relaxed SCA approach for calibrating low rank correlation matrix problem

Abstract

In this paper, we consider a class of rank constrained correlation matrix calibrating problems with simple upper and lower bounds. This problem can be reformulated into a DC (difference of convex) constrained problem, thus sequential convex approximation (SCA) type approaches can be considered. However, classical SCA approach cannot be directly applied since the constraint qualification needed in convergence theorem does not hold for this problem. This motivates us to develop a relaxed SCA approach. We prove that for relaxed DC problems, the sets of their stationary points converge to the set of stationary points of the original problem as the relaxation parameter approaching to zero. For each relaxed DC problem, we apply the SCA approach to generate a sequence of convex subproblems. We show that all cluster points of optimal solutions of these subproblems are stationary points of the relaxed DC problem. Numerical results verify the efficiency of our approach.