A difference-of-convex approach for split feasibility with applications to matrix factorizations and outlier detection

Chen Chen, Ting Kei Pong, Lulin Tan, Liaoyuan Zeng

Research output: Journal article publicationJournal articleAcademic researchpeer-review

2 Citations (Scopus)

Abstract

The split feasibility problem is to find an element in the intersection of a closed set C and the linear preimage of another closed set D, assuming the projections onto C and D are easy to compute. This class of problems arises naturally in many contemporary applications such as compressed sensing. While the sets C and D are typically assumed to be convex in the literature, in this paper, we allow both sets to be possibly nonconvex. We observe that, in this setting, the split feasibility problem can be formulated as an optimization problem with a difference-of-convex objective so that standard majorization-minimization type algorithms can be applied. Here we focus on the nonmonotone proximal gradient algorithm with majorization studied in Liu et al. (Math Program, 2019. https://doi.org/10.1007/s10107-018-1327-8, Appendix A). We show that, when this algorithm is applied to a split feasibility problem, the sequence generated clusters at a stationary point of the problem under mild assumptions. We also study local convergence property of the sequence under suitable assumptions on the closed sets involved. Finally, we perform numerical experiments to illustrate the efficiency of our approach on solving split feasibility problems that arise in completely positive matrix factorization, (uniformly) sparse matrix factorization, and outlier detection.

Original languageEnglish
Pages (from-to)107-136
Number of pages30
JournalJournal of Global Optimization
Volume78
Issue number1
DOIs
Publication statusPublished - 1 Sep 2020

Keywords

  • Difference-of-convex optimization
  • Matrix factorizations
  • Split feasibility problems

ASJC Scopus subject areas

  • Computer Science Applications
  • Management Science and Operations Research
  • Control and Optimization
  • Applied Mathematics

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