Variable Smoothing for Weakly Convex Composite Functions

2020/03/31 by Axel Böhm, Stephen J. Wright · 5 citations
Mathematics · #math.OC

paper · pdf · doi:10.1007/s10915-020-01332-8

arxiv created 2021/05/31 · arxiv updated 2021/06/01

Abstract

We study minimization of a structured objective function, being the sum of a smooth function and a composition of a weakly convex function with a linear operator. Applications include image reconstruction problems with regularizers that introduce less bias than the standard convex regularizers. We develop a variable smoothing algorithm, based on the Moreau envelope with a decreasing sequence of smoothing parameters, and prove a complexity of O(ε^-3) to achieve an ε-approximate solution. This bound interpolates between the O(ε^-2) bound for the smooth case and the O(ε^-4) bound for the subgradient method. Our complexity bound is in line with other works that deal with structured nonsmoothness of weakly convex functions.

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