Please use this identifier to cite or link to this item: http://hdl.handle.net/10397/66695
PIRA download icon_1.1View/Download Full Text
Title: Douglas-Rachford splitting for nonconvex optimization with application to nonconvex feasibility problems
Authors: Li, G
Pong, TK 
Issue Date: Sep-2016
Source: Mathematical programming, Sept. 2016, v. 159, no. 1-2, p. 371-401
Abstract: We adapt the Douglas–Rachford (DR) splitting method to solve nonconvex feasibility problems by studying this method for a class of nonconvex optimization problem. While the convergence properties of the method for convex problems have been well studied, far less is known in the nonconvex setting. In this paper, for the direct adaptation of the method to minimize the sum of a proper closed function g and a smooth function f with a Lipschitz continuous gradient,we showthat if the step-size parameter is smaller than a computable threshold and the sequence generated has a cluster point, then it gives a stationary point of the optimization problem. Convergence of the whole sequence and a local convergence rate are also established under the additional assumption that f and g are semi-algebraic.We also give simple sufficient conditions guaranteeing the boundedness of the sequence generated. We then apply our nonconvex DR splitting method to finding a point in the intersection of a closed convex set C and a general closed set D by minimizing the squared distance to C subject to D. We show that if either set is bounded and the step-size parameter is smaller than a computable threshold, then the sequence generated from theDRsplitting method is actually bounded. Consequently, the sequence generated will have cluster points that are stationary for an optimization problem, and the whole sequence is convergent under an additional assumption that C and D are semi-algebraic. We achieve these results based on a new merit function constructed particularly for the DR splitting method. Our preliminary numerical results indicate that our DR splitting method usually outperforms the alternating projection method in finding a sparse solution of a linear system, in terms of both the solution quality and the number of iterations taken.
Publisher: Springer
Journal: Mathematical programming 
ISSN: 0025-5610
DOI: 10.1007/s10107-015-0963-5
Rights: © Springer-Verlag Berlin Heidelberg and Mathematical Optimization Society 2015
This version of the article has been accepted for publication, after peer review (when applicable) and is subject to Springer Nature’s AM terms of use (https://www.springernature.com/gp/open-research/policies/accepted-manuscript-terms), but is not the Version of Record and does not reflect post-acceptance improvements, or any corrections. The Version of Record is available online at: http://dx.doi.org/10.1007/s10107-015-0963-5
Appears in Collections:Journal/Magazine Article

Files in This Item:
File Description SizeFormat 
Pong_Douglas–Rachford_Splitting_Nonconvex.pdfPre-Published version1 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

207
Last Week
3
Last month
Citations as of Aug 7, 2022

Downloads

6
Citations as of Aug 7, 2022

SCOPUSTM   
Citations

72
Last Week
0
Last month
Citations as of Aug 4, 2022

WEB OF SCIENCETM
Citations

72
Last Week
0
Last month
Citations as of Aug 4, 2022

Google ScholarTM

Check

Altmetric


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