Variable Smoothing for Weakly Convex Composite Functions

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