Further properties of the forward-backward envelope with applications to difference-of-convex programming

Abstract

In this paper, we further study the forward-backward envelope first introduced in Patrinos and Bemporad (Proceedings of the IEEE Conference on Decision and Control, pp 2358-2363, 2013) and Stella et al. (Comput Optim Appl, doi:10.1007/s10589-017-9912-y, 2017) for problems whose objective is the sum of a proper closed convex function and a twice continuously differentiable possibly nonconvex function with Lipschitz continuous gradient. We derive sufficient conditions on the original problem for the corresponding forward-backward envelope to be a level-bounded and Kurdyka-Aojasiewicz function with an exponent of ; these results are important for the efficient minimization of the forward-backward envelope by classical optimization algorithms. In addition, we demonstrate how to minimize some difference-of-convex regularized least squares problems by minimizing a suitably constructed forward-backward envelope. Our preliminary numerical results on randomly generated instances of large-scale regularized least squares problems (Yin et al. in SIAM J Sci Comput 37:A536-A563, 2015) illustrate that an implementation of this approach with a limited-memory BFGS scheme usually outperforms standard first-order methods such as the nonmonotone proximal gradient method in Wright et al. (IEEE Trans Signal Process 57:2479-2493, 2009).