A global exact penalty for rank-constrained optimization problem and applications

This paper considers a rank-constrained optimization problem where the objective function is continuously differentiable on a closed convex set. After replacing the rank constraint by an equality of the truncated difference of L1 and L2 norm, and adding the equality constraint into the objective to get a penalty problem, we prove that the penalty problem is exact in the sense that the set of its global (local) optimal solutions coincides with that of the original problem when the penalty parameter is over a certain threshold. This establishes the theoretical guarantee for the truncated difference of L1 and L2 norm regularization optimization including the work of Ma et al. (SIAM J Imaging Sci 10(3):1346-1380, 2017). Besides, for the penalty problem, we propose an extrapolation proximal difference of convex algorithm (epDCA) and prove the sequence generated by epDCA converges to a stationary point of the penalty problem. Further, an adaptive penalty method based on epDCA is constructed for the original rank-constrained problem. The efficiency of the algorithms is verified via numerical experiments for the nearest low-rank correlation matrix problem and the matrix completion problem.