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Variable Smoothing for Weakly Convex Composite Functions

Author

Listed:
  • Axel Böhm

    (University of Vienna)

  • Stephen J. Wright

    (University of Wisconsin-Madison)

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.

Suggested Citation

  • Axel Böhm & Stephen J. Wright, 2021. "Variable Smoothing for Weakly Convex Composite Functions," Journal of Optimization Theory and Applications, Springer, vol. 188(3), pages 628-649, March.
    • Handle: RePEc:spr:joptap:v:188:y:2021:i:3:d:10.1007_s10957-020-01800-z
    DOI: 10.1007/s10957-020-01800-z
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    References listed on IDEAS

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    1. Radu Boţ & Christopher Hendrich, 2015. "A variable smoothing algorithm for solving convex optimization problems," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 23(1), pages 124-150, April.
    2. Jianqing Fan, 1997. "Comments on «Wavelets in statistics: A review» by A. Antoniadis," Statistical Methods & Applications, Springer;Società Italiana di Statistica, vol. 6(2), pages 131-138, August.
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