Variable Smoothing for Weakly Convex Composite Functions

Axel Böhm () and Stephen J. Wright ()
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Axel Böhm: University of Vienna
Stephen J. Wright: University of Wisconsin-Madison

Journal of Optimization Theory and Applications, 2021, vol. 188, issue 3, No 2, 628-649

Abstract: 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.

Keywords: Variable smoothing; Weakly convex; Composite minimization (search for similar items in EconPapers)
Date: 2021
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DOI: 10.1007/s10957-020-01800-z

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