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Title: A subgradient-based approach for finding the maximum feasible subsystem with respect to a set
Authors: Ye, M
Pong, TK 
Issue Date: 2020
Source: SIAM journal on optimization, 2020, v. 30, no. 2, p. 1274-1299
Abstract: We propose a subgradient-based method for finding the maximum feasible subsystem in a collection of closed sets with respect to a given closed set C (MFSC). In this method, we reformulate the MFSC problem as an ℓ0 optimization problem and construct a sequence of continuous optimization problems to approximate it. The objective of each approximation problem is the sum of the composition of a nonnegative nondecreasing continuously differentiable concave function with the squared distance function to a closed set. Although this objective function is nonsmooth in general, a subgradient can be obtained in terms of the projections onto the closed sets. Based on this observation, we adapt a subgradient projection method to solve these approximation problems. Unlike classical subgradient methods, the convergence (clustering to stationary points) of our subgradient method is guaranteed with a nondiminishing stepsize under mild assumptions. This allows us to further study the sequential convergence of the subgradient method under suitable Kurdyka-Łojasiewicz assumptions. Finally, we illustrate our algorithm numerically for solving the MFSC problems on a collection of halfspaces and a collection of unions of halfspaces, respectively, with respect to the set of s-sparse vectors.
Keywords: Kurdyka-Łojasiewicz property
Maximum feasible subsystem
Subgradient methods
Publisher: Society for Industrial and Applied Mathematics
Journal: SIAM journal on optimization 
ISSN: 1052-6234
EISSN: 1095-7189
DOI: 10.1137/18M1186320
Rights: © 2020 Society for Industrial and Applied Mathematics
First Published in SIAM Journal on Optimization in Volume 30, Issue 2, published by the Society for Industrial and Applied Mathematics (SIAM). Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.
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