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Title: Linear convergence of proximal gradient algorithm with extrapolation for a class of nonconvex nonsmooth minimization problems
Authors: Wen, B 
Chen, X 
Pong, TK 
Issue Date: 2017
Source: SIAM journal on optimization, 2017, v. 27, no. 1, p. 124-145
Abstract: In this paper, we study the proximal gradient algorithm with extrapolation for minimizing the sum of a Lipschitz differentiable function and a proper closed convex function. Under the error bound condition used in [Ann. Oper. Res., 46 (1993), pp. 157-178] for analyzing the convergence of the proximal gradient algorithm, we show that there exists a threshold such that if the extrapolation coefficients are chosen below this threshold, then the sequence generated converges R-linearly to a stationary point of the problem. Moreover, the corresponding sequence of objective values is also R-linearly convergent. In addition, the threshold reduces to 1 for convex problems, and as a consequence we obtain the R-linear convergence of the sequence generated by FISTA with fixed restart. Finally, we present some numerical experiments to illustrate our results.
Keywords: Linear convergence
Extrapolation
Error bound
Accelerated gradient method
Nonconvex nonsmooth minimization
Convex minimization
Publisher: Society for Industrial and Applied Mathematics
Journal: SIAM journal on optimization 
ISSN: 1052-6234
EISSN: 1095-7189
DOI: 10.1137/16M1055323
Rights: © 2017 Society for Industrial and Applied Mathematics
The following publication Wen, B., Chen, X., & Pong, T. K. (2017). Linear convergence of proximal gradient algorithm with extrapolation for a class of nonconvex nonsmooth minimization problems. SIAM Journal on Optimization, 27(1), 124-145 is available at https://doi.org/10.1137/16M1055323
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