Please use this identifier to cite or link to this item: http://hdl.handle.net/10397/7015
Title: Worst-case complexity of smoothing quadratic regularization methods for non-lipschitzian optimization
Authors: Bian, W
Chen, X 
Keywords: Nonsmooth nonconvex optimization
Smoothing approximation
Quadratic regularization
Convergence
Worst-case complexity
Stationary point
Issue Date: 2013
Publisher: Society for Industrial and Applied Mathematics
Source: SIAM Journal on optimization, 2013, v. 23, no. 3, p. 1718–1741 How to cite?
Journal: SIAM Journal on optimization 
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.
URI: http://hdl.handle.net/10397/7015
ISSN: 1052-6234 (print)
1095-7189 (online)
DOI: 10.1137/120864908
Rights: © 2013 Society for Industrial and Applied Mathematics
Appears in Collections:Journal/Magazine Article

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