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Title: Further properties of the forward-backward envelope with applications to difference-of-convex programming
Authors: Liu, TX 
Pong, TK 
Keywords: Forward-backward envelope
Kurdyka-Lojasiewicz property
Difference-of-convex programming
Issue Date: 2017
Publisher: Springer
Source: Computational optimization and applications, 2017, v. 67, no. 3, p. 489-520 How to cite?
Journal: Computational optimization and applications 
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).
ISSN: 0926-6003
EISSN: 1573-2894
DOI: 10.1007/s10589-017-9900-2
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