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

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...

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Veröffentlicht in:Computational optimization and applications Jg. 67; H. 3; S. 489 - 520
Hauptverfasser: Liu, Tianxiang, Pong, Ting Kei
Format: Journal Article
Sprache:Englisch
Veröffentlicht: New York Springer US 01.07.2017
Springer Nature B.V
Schlagworte:
Computational geometry
Continuity (mathematics)
Convex and Discrete Geometry
Convexity
Economic models
Least squares method
Management Science
Mathematical programming
Mathematics
Mathematics and Statistics
Operations Research
Operations Research/Decision Theory
Optimization
Optimization algorithms
Signal processing
Statistics
Kurdyka–Łojasiewicz property
Forward–backward envelope
Difference-of-convex programming
ISSN:0926-6003, 1573-2894
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Zusammenfassung: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 1 2 ; 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 ℓ 1 - 2 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 ).
Bibliographie:ObjectType-Article-1
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ISSN:0926-6003
1573-2894
DOI:10.1007/s10589-017-9900-2