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Mirror Prox algorithm for multi-term composite minimization and semi-separable problems

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  • Niao He
  • Anatoli Juditsky
  • Arkadi Nemirovski

Abstract

In the paper, we develop a composite version of Mirror Prox algorithm for solving convex–concave saddle point problems and monotone variational inequalities of special structure, allowing to cover saddle point/variational analogies of what is usually called “composite minimization” (minimizing a sum of an easy-to-handle nonsmooth and a general-type smooth convex functions “as if” there were no nonsmooth component at all). We demonstrate that the composite Mirror Prox inherits the favourable (and unimprovable already in the large-scale bilinear saddle point case) [InlineEquation not available: see fulltext.] efficiency estimate of its prototype. We demonstrate that the proposed approach can be successfully applied to Lasso-type problems with several penalizing terms (e.g. acting together $$\ell _1$$ ℓ 1 and nuclear norm regularization) and to problems of semi-separable structures considered in the alternating directions methods, implying in both cases methods with the [InlineEquation not available: see fulltext.] complexity bounds. Copyright Springer Science+Business Media New York 2015

Suggested Citation

  • Niao He & Anatoli Juditsky & Arkadi Nemirovski, 2015. "Mirror Prox algorithm for multi-term composite minimization and semi-separable problems," Computational Optimization and Applications, Springer, vol. 61(2), pages 275-319, June.
    • Handle: RePEc:spr:coopap:v:61:y:2015:i:2:p:275-319
    DOI: 10.1007/s10589-014-9723-3
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    References listed on IDEAS

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    1. NESTEROV, Yurii, 2013. "Gradient methods for minimizing composite functions," LIDAM Reprints CORE 2510, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    2. Arkadi Nemirovski & Reuven Y. Rubinstein, 2002. "An Efficient Stochastic Approximation Algorithm for Stochastic Saddle Point Problems," International Series in Operations Research & Management Science, in: Moshe Dror & Pierre L’Ecuyer & Ferenc Szidarovszky (ed.), Modeling Uncertainty, chapter 0, pages 156-184, Springer.
    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).
    4. Arkadi Nemirovski & Shmuel Onn & Uriel G. Rothblum, 2010. "Accuracy Certificates for Computational Problems with Convex Structure," Mathematics of Operations Research, INFORMS, vol. 35(1), pages 52-78, February.
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    Cited by:

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    2. 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.
    3. Jueyou Li & Guo Chen & Zhaoyang Dong & Zhiyou Wu, 2016. "A fast dual proximal-gradient method for separable convex optimization with linear coupled constraints," Computational Optimization and Applications, Springer, vol. 64(3), pages 671-697, July.

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