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A variable smoothing algorithm for solving convex optimization problems

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  • Radu Boţ
  • Christopher Hendrich

Abstract

In this article, we propose a method for solving unconstrained optimization problems with convex and Lipschitz continuous objective functions. By making use of the Moreau envelopes of the functions occurring in the objective, we smooth them to convex and differentiable functions with Lipschitz continuous gradients using both variable and constant smoothing parameters. The resulting problem is solved via an accelerated first-order method and this allows us to recover approximately the optimal solutions to the initial optimization problem with a rate of convergence of order $$\mathcal {O}\left( \tfrac{\ln k}{k}\right) $$ O ln k k for variable smoothing and of order $$\mathcal {O}\left( \tfrac{1}{k}\right) $$ O 1 k for constant smoothing. Some numerical experiments employing the variable smoothing method in image processing and in supervised learning classification are also presented. Copyright Sociedad de Estadística e Investigación Operativa 2015

Suggested Citation

  • 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.
    • Handle: RePEc:spr:topjnl:v:23:y:2015:i:1:p:124-150
    DOI: 10.1007/s11750-014-0326-z
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    References listed on IDEAS

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    1. Radu Boţ & Christopher Hendrich, 2013. "A double smoothing technique for solving unconstrained nondifferentiable convex optimization problems," Computational Optimization and Applications, Springer, vol. 54(2), pages 239-262, March.
    2. DEVOLDER, Olivier & GLINEUR, François & NESTEROV, Yurii, 2012. "Double smoothing technique for large-scale linearly constrained convex optimization," LIDAM Reprints CORE 2423, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    3. Bot, Radu Ioan & Lorenz, Nicole, 2011. "Optimization problems in statistical learning: Duality and optimality conditions," European Journal of Operational Research, Elsevier, vol. 213(2), pages 395-404, September.
    4. Patrick L. Combettes & Jean-Christophe Pesquet, 2011. "Proximal Splitting Methods in Signal Processing," Springer Optimization and Its Applications, in: Heinz H. Bauschke & Regina S. Burachik & Patrick L. Combettes & Veit Elser & D. Russell Luke & Henry (ed.), Fixed-Point Algorithms for Inverse Problems in Science and Engineering, chapter 0, pages 185-212, Springer.
    5. NESTEROV, Yu., 2005. "Smooth minimization of non-smooth functions," LIDAM Reprints CORE 1819, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
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    Cited by:

    1. Fan Wu & Wei Bian, 2023. "Smoothing Accelerated Proximal Gradient Method with Fast Convergence Rate for Nonsmooth Convex Optimization Beyond Differentiability," Journal of Optimization Theory and Applications, Springer, vol. 197(2), pages 539-572, May.
    2. 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.

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