Linear Convergence of Proximal Gradient Algorithm with Extrapolation for a Class of Nonconvex Nonsmooth Minimization Problems
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 [19] 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 -linearly to a stationary point of the problem. Moreover, the corresponding sequence of objective values is also -linearly convergent. In addition, the threshold reduces to for convex problems and, as a consequence, we obtain the -linear convergence of the sequence generated by FISTA with fixed restart. Finally, we present some numerical experiments to illustrate our results.
Cite
@article{arxiv.1512.09302,
title = {Linear Convergence of Proximal Gradient Algorithm with Extrapolation for a Class of Nonconvex Nonsmooth Minimization Problems},
author = {Bo Wen and Xiaojun Chen and Ting Kei Pong},
journal= {arXiv preprint arXiv:1512.09302},
year = {2016}
}
Comments
We have replaced the blanket assumptions on $f+g$ by the (weaker) assumptions that the optimal value of (1.1) is finite and attained. Section 3.4 has been deleted