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Decomposition Techniques for Bilinear Saddle Point Problems and Variational Inequalities with Affine Monotone Operators

Author

Listed:
  • Bruce Cox

    (US Air Force)

  • Anatoli Juditsky

    (Université Grenoble Alpes)

  • Arkadi Nemirovski

    (Georgia Institute of Technology)

Abstract

The majority of first-order methods for large-scale convex–concave saddle point problems and variational inequalities with monotone operators are proximal algorithms. To make such an algorithm practical, the problem’s domain should be proximal-friendly—admit a strongly convex function with easy to minimize linear perturbations. As a by-product, this domain admits a computationally cheap linear minimization oracle (LMO) capable to minimize linear forms. There are, however, important situations where a cheap LMO indeed is available, but the problem domain is not proximal-friendly, which motivates search for algorithms based solely on LMO. For smooth convex minimization, there exists a classical algorithm using LMO—conditional gradient. In contrast, known to us similar techniques for other problems with convex structure (nonsmooth convex minimization, convex–concave saddle point problems, even as simple as bilinear ones, and variational inequalities with monotone operators, even as simple as affine) are quite recent and utilize common approach based on Fenchel-type representations of the associated objectives/vector fields. The goal of this paper was to develop alternative (and seemingly much simpler) decomposition techniques based on LMO for bilinear saddle point problems and for variational inequalities with affine monotone operators.

Suggested Citation

  • Bruce Cox & Anatoli Juditsky & Arkadi Nemirovski, 2017. "Decomposition Techniques for Bilinear Saddle Point Problems and Variational Inequalities with Affine Monotone Operators," Journal of Optimization Theory and Applications, Springer, vol. 172(2), pages 402-435, February.
  • Handle: RePEc:spr:joptap:v:172:y:2017:i:2:d:10.1007_s10957-016-0949-3
    DOI: 10.1007/s10957-016-0949-3
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    References listed on IDEAS

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    1. Pavel Dvurechensky & Yurii Nesterov & Vladimir Spokoiny, 2015. "Primal-Dual Methods for Solving Infinite-Dimensional Games," Journal of Optimization Theory and Applications, Springer, vol. 166(1), pages 23-51, July.
    2. DVURECHENSKY, Pavel & NESTEROV, Yurii & SPOKOINY, Vladimir, 2015. "Primal-dual methods for solving infinite-dimensional games," LIDAM Reprints CORE 2700, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    3. 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:

    1. Guangming Zhou & Qin Wang & Wenjie Zhao, 2020. "Saddle points of rational functions," Computational Optimization and Applications, Springer, vol. 75(3), pages 817-832, April.
    2. Wenjie Zhao & Guangming Zhou, 2022. "Local saddle points for unconstrained polynomial optimization," Computational Optimization and Applications, Springer, vol. 82(1), pages 89-106, May.

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