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A double smoothing technique for solving unconstrained nondifferentiable convex optimization problems

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  • Radu Boţ
  • Christopher Hendrich

Abstract

The aim of this paper is to develop an efficient algorithm for solving a class of unconstrained nondifferentiable convex optimization problems in finite dimensional spaces. To this end we formulate first its Fenchel dual problem and regularize it in two steps into a differentiable strongly convex one with Lipschitz continuous gradient. The doubly regularized dual problem is then solved via a fast gradient method with the aim of accelerating the resulting convergence scheme. The theoretical results are finally applied to an l 1 regularization problem arising in image processing. Copyright Springer Science+Business Media New York 2013

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  • 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.
    • Handle: RePEc:spr:coopap:v:54:y:2013:i:2:p:239-262
    DOI: 10.1007/s10589-012-9523-6
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    References listed on IDEAS

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    1. 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).
    2. repec:cor:louvrp:-2423 is not listed on IDEAS
    3. 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. 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. Masoud Ahookhosh & Arnold Neumaier, 2018. "Solving structured nonsmooth convex optimization with complexity $$\mathcal {O}(\varepsilon ^{-1/2})$$ O ( ε - 1 / 2 )," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 26(1), pages 110-145, April.
    3. E. M. Bednarczuk & A. Jezierska & K. E. Rutkowski, 2018. "Proximal primal–dual best approximation algorithm with memory," Computational Optimization and Applications, Springer, vol. 71(3), pages 767-794, December.
    4. Gert Wanka & Oleg Wilfer, 2017. "A Lagrange duality approach for multi-composed optimization problems," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 25(2), pages 288-313, July.
    5. Quoc Tran-Dinh, 2017. "Adaptive smoothing algorithms for nonsmooth composite convex minimization," Computational Optimization and Applications, Springer, vol. 66(3), pages 425-451, April.

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