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Title: Penalty methods for a class of non-Lipschitz optimization problems
Authors: Chen, X 
Lu, Z
Pong, TK 
Issue Date: 2016
Source: SIAM journal on optimization, 2016, v. 26, no. 3, p. 1465-1492
Abstract: We consider a class of constrained optimization problems with a possibly nonconvex non-Lipschitz objective and a convex feasible set being the intersection of a polyhedron and a possibly degenerate ellipsoid. Such problems have a wide range of applications in data science, where the objective is used for inducing sparsity in the solutions while the constraint set models the noise tolerance and incorporates other prior information for data fitting. To solve this class of constrained optimization problems, a common approach is the penalty method. However, there is little theory on exact penalization for problems with nonconvex and non-Lipschitz objective functions. In this paper, we study the existence of exact penalty parameters regarding local minimizers, stationary points, and $\epsilon$-minimizers under suitable assumptions. Moreover, we discuss a penalty method whose subproblems are solved via a nonmonotone proximal gradient method with a suitable update scheme for the penalty parameters and prove the convergence of the algorithm to a KKT point of the constrained problem. Preliminary numerical results demonstrate the efficiency of the penalty method for finding sparse solutions of underdetermined linear systems.
Keywords: Exact penalty
Proximal gradient method
Sparse solution
Nonconvex optimization
Non-Lipschitz optimization
Publisher: Society for Industrial and Applied Mathematics
Journal: SIAM journal on optimization 
ISSN: 1052-6234
EISSN: 1095-7189
DOI: 10.1137/15M1028054
Rights: © 2016 Society for Industrial and Applied Mathematics
The following publication Chen, X., Lu, Z., & Pong, T. K. (2016). Penalty methods for a class of non-Lipschitz optimization problems. SIAM Journal on Optimization, 26(3), 1465-1492 is available at is available at
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