Please use this identifier to cite or link to this item: http://hdl.handle.net/10397/89216
PIRA download icon_1.1View/Download Full Text
Title: A successive difference-of-convex approximation method for a class of nonconvex nonsmooth optimization problems
Authors: Liu, T 
Pong, TK 
Takeda, A
Issue Date: Jul-2019
Source: Mathematical programming, July 2019, v. 176, p. 339-367
Abstract: We consider a class of nonconvex nonsmooth optimization problems whose objective is the sum of a smooth function and a finite number of nonnegative proper closed possibly nonsmooth functions (whose proximal mappings are easy to compute), some of which are further composed with linear maps. This kind of problems arises naturally in various applications when different regularizers are introduced for inducing simultaneous structures in the solutions. Solving these problems, however, can be challenging because of the coupled nonsmooth functions: the corresponding proximal mapping can be hard to compute so that standard first-order methods such as the proximal gradient algorithm cannot be applied efficiently. In this paper, we propose a successive difference-of-convex approximation method for solving this kind of problems. In this algorithm, we approximate the nonsmooth functions by their Moreau envelopes in each iteration. Making use of the simple observation that Moreau envelopes of nonnegative proper closed functions are continuous difference-of-convex functions, we can then approximately minimize the approximation function by first-order methods with suitable majorization techniques. These first-order methods can be implemented efficiently thanks to the fact that the proximal mapping of each nonsmooth function is easy to compute. Under suitable assumptions, we prove that the sequence generated by our method is bounded and any accumulation point is a stationary point of the objective. We also discuss how our method can be applied to concrete applications such as nonconvex fused regularized optimization problems and simultaneously structured matrix optimization problems, and illustrate the performance numerically for these two specific applications.
Keywords: Difference-of-convex approximation
Moreau envelope
Proximal mapping
Simultaneous structures
Publisher: Springer
Journal: Mathematical programming 
ISSN: 0025-5610
DOI: 10.1007/s10107-018-1327-8
Rights: © Springer-Verlag GmbH Germany, part of Springer Nature and Mathematical Optimization Society 2018
This is a post-peer-review, pre-copyedit version of an article published in Mathematical Programming. The final authenticated version is available online at: http://dx.doi.org/10.1007/s10107-018-1327-8
Appears in Collections:Journal/Magazine Article

Files in This Item:
File Description SizeFormat 
a0585-n05_SDCAM_re3_updated.pdfPre-Published version1.09 MBAdobe PDFView/Open
Open Access Information
Status open access
File Version Final Accepted Manuscript
Access
View full-text via PolyU eLinks SFX Query
Show full item record

Page views

88
Last Week
8
Last month
Citations as of Mar 24, 2024

Downloads

25
Citations as of Mar 24, 2024

SCOPUSTM   
Citations

21
Citations as of Mar 28, 2024

WEB OF SCIENCETM
Citations

19
Citations as of Mar 28, 2024

Google ScholarTM

Check

Altmetric


Items in DSpace are protected by copyright, with all rights reserved, unless otherwise indicated.