A novel method for a class of structured low-rank minimizations with equality constraint

The positive semidefinite constraint and equality constraint arise widely in matrix optimization problems of different areas including signal/image processing, finance and risk management. In this paper, an inexact accelerated Augmented Lagrangian Method (ALM) relying on a parameter m is designed to solve the structured low-rank minimization with equality constraint, which is more general and flexible than the existing ALM and its variants. We prove a worst-case O(1∕k²) convergence rate of the new method in terms of the residual of the Lagrangian function, and we analyze that when m∈[0,1) the residual of our method is smaller than that of the traditional accelerated ALM. Compared with several state-of-the-art methods, preliminary numerical experiments on solving the Q-weighted low-rank correlation matrix problem from finance validate the efficiency of the proposed method.