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Descentwise inexact proximal algorithms for smooth optimization

Author

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  • Marc Fuentes
  • Jérôme Malick

  • Claude Lemaréchal

Abstract

The proximal method is a standard regularization approach in optimization. Practical implementations of this algorithm require (i) an algorithm to compute the proximal point, (ii) a rule to stop this algorithm, (iii) an update formula for the proximal parameter. In this work we focus on (ii), when smoothness is present—so that Newton-like methods can be used for (i): we aim at giving adequate stopping rules to reach overall efficiency of the method. Roughly speaking, usual rules consist in stopping inner iterations when the current iterate is close to the proximal point. By contrast, we use the standard paradigm of numerical optimization: the basis for our stopping test is a “sufficient” decrease of the objective function, namely a fraction of the ideal decrease. We establish convergence of the algorithm thus obtained and we illustrate it on some ill-conditioned problems. The experiments show that combining the proposed inexact proximal scheme with a standard smooth optimization algorithm improves the numerical behaviour of the latter for those ill-conditioned problems. Copyright Springer Science+Business Media, LLC 2012

Suggested Citation

  • Marc Fuentes & Jérôme Malick & Claude Lemaréchal, 2012. "Descentwise inexact proximal algorithms for smooth optimization," Computational Optimization and Applications, Springer, vol. 53(3), pages 755-769, December.
    • Handle: RePEc:spr:coopap:v:53:y:2012:i:3:p:755-769
    DOI: 10.1007/s10589-012-9461-3
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    References listed on IDEAS

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    1. R. T. Rockafellar, 1976. "Augmented Lagrangians and Applications of the Proximal Point Algorithm in Convex Programming," Mathematics of Operations Research, INFORMS, vol. 1(2), pages 97-116, May.
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    1. Paul Armand & Ngoc Nguyen Tran, 2021. "Local Convergence Analysis of a Primal–Dual Method for Bound-Constrained Optimization Without SOSC," Journal of Optimization Theory and Applications, Springer, vol. 189(1), pages 96-116, April.
    2. Glaydston Carvalho Bento & João Xavier Cruz Neto & Antoine Soubeyran & Valdinês Leite Sousa Júnior, 2016. "Dual Descent Methods as Tension Reduction Systems," Journal of Optimization Theory and Applications, Springer, vol. 171(1), pages 209-227, October.
    3. Franck Iutzeler & Jérôme Malick, 2018. "On the Proximal Gradient Algorithm with Alternated Inertia," Journal of Optimization Theory and Applications, Springer, vol. 176(3), pages 688-710, March.
    4. Elizabeth Karas & Sandra Santos & Benar Svaiter, 2015. "Algebraic rules for quadratic regularization of Newton’s method," Computational Optimization and Applications, Springer, vol. 60(2), pages 343-376, March.
    5. Paul Armand & Isaï Lankoandé, 2017. "An inexact proximal regularization method for unconstrained optimization," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 85(1), pages 43-59, February.

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