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Title: A difference-of-convex approach for split feasibility with applications to matrix factorizations and outlier detection
Authors: Chen, C
Pong, TK 
Tan, L
Zeng, L 
Issue Date: Sep-2020
Source: Journal of global optimization, Sept. 2020, v. 78, no. 1, p. 107-136
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.
Keywords: Difference-of-convex optimization
Matrix factorizations
Split feasibility problems
Publisher: Springer
Journal: Journal of global optimization 
ISSN: 0925-5001
EISSN: 1573-2916
DOI: 10.1007/s10898-020-00899-8
Rights: © Springer Science+Business Media, LLC, part of Springer Nature 2020
This is a post-peer-review, pre-copyedit version of an article published in Journal of Global Optimization. The final authenticated version is available online at: https://dx.doi.org/10.1007/s10898-020-00899-8.
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