Variable Smoothing for Weakly Convex Composite Functions
Axel Böhm, Stephen J. Wright · Journal of Optimization Theory and Applications · 2021
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 $${\mathcal {O}}(\epsilon ^{-3})$$ O(ϵ-3) to achieve an $$\epsilon $$ ϵ -approximate solution. This bound interpolates between the $${\mathcal {O}}(\epsilon ^{-2})$$ O(ϵ-2) bound for the smooth case and the $${\mathcal {O}}(\epsilon ^{-4})$$ 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.