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http://hdl.handle.net/10397/66842
| 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 https://doi.org/10.1137/15M1028054 |
| Appears in Collections: | Journal/Magazine Article |
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