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http://hdl.handle.net/10397/7015
Title: | Worst-case complexity of smoothing quadratic regularization methods for non-lipschitzian optimization | Authors: | Bian, W Chen, X |
Issue Date: | 2013 | Source: | SIAM Journal on optimization, 2013, v. 23, no. 3, p. 1718–1741 | Abstract: | In this paper, we propose a smoothing quadratic regularization (SQR) algorithm for solving a class of nonsmooth nonconvex, perhaps even non-Lipschitzian minimization problems, which has wide applications in statistics and sparse reconstruction. The proposed SQR algorithm is a first order method. At each iteration, the SQR algorithm solves a strongly convex quadratic minimization problem with a diagonal Hessian matrix, which has a simple closed-form solution that is inexpensive to calculate. We show that the worst-case complexity of reaching an ϵ scaled stationary point is $O(ϵ⁻²). Moreover, if the objective function is locally Lipschitz continuous, the SQR algorithm with a slightly modified updating scheme for the smoothing parameter and iterate can obtain an ϵ Clarke stationary point in at most $O(ϵ⁻³) iterations. | Keywords: | Nonsmooth nonconvex optimization Smoothing approximation Quadratic regularization Convergence Worst-case complexity Stationary point |
Publisher: | Society for Industrial and Applied Mathematics | Journal: | SIAM Journal on optimization | ISSN: | 1052-6234 | EISSN: | 1095-7189 | DOI: | 10.1137/120864908 | Rights: | © 2013 Society for Industrial and Applied Mathematics |
Appears in Collections: | Journal/Magazine Article |
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Bian_Worst-case_Complexity_quadratic.pdf | 579.7 kB | Adobe PDF | View/Open |
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