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

Tianxiang Liu, Ting Kei Pong

Research output: Journal article publicationJournal articleAcademic researchpeer-review

17 Citations (Scopus)

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–Łojasiewicz function with an exponent of 12; 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 ℓ1 - 2regularized 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).
Original languageEnglish
Pages (from-to)489-520
Number of pages32
JournalComputational Optimization and Applications
Volume67
Issue number3
DOIs
Publication statusPublished - 1 Jul 2017

Keywords

  • Difference-of-convex programming
  • Forward–backward envelope
  • Kurdyka–Łojasiewicz property

ASJC Scopus subject areas

  • Control and Optimization
  • Computational Mathematics
  • Applied Mathematics

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