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Tight Ergodic Sublinear Convergence Rate of the Relaxed Proximal Point Algorithm for Monotone Variational Inequalities

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  • Guoyong Gu

    (Nanjing University)

  • Junfeng Yang

    (Nanjing University)

Abstract

This paper considers the relaxed proximal point algorithm for solving monotone variational inequality problems, and our main contribution is the establishment of a tight ergodic sublinear convergence rate. First, the tight or exact worst-case convergence rate is computed using the performance estimation framework. It is observed that this numerical bound asymptotically coincides with the best-known existing rate, whose tightness is not clear. This implies that, without further assumptions, sublinear convergence rate is likely the best achievable rate for the relaxed proximal point algorithm. Motivated by the numerical result, a concrete example is constructed, which provides a lower bound on the exact worst-case convergence rate. This lower bound coincides with the numerical bound computed via the performance estimation framework, leading us to conjecture that the lower bound provided by the example is exactly the tight worse-case rate, which is then verified theoretically. We thus have established an ergodic sublinear complexity rate that is tight in terms of both the sublinear order and the constants involved.

Suggested Citation

  • Guoyong Gu & Junfeng Yang, 2024. "Tight Ergodic Sublinear Convergence Rate of the Relaxed Proximal Point Algorithm for Monotone Variational Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 202(1), pages 373-387, July.
    • Handle: RePEc:spr:joptap:v:202:y:2024:i:1:d:10.1007_s10957-022-02058-3
    DOI: 10.1007/s10957-022-02058-3
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    References listed on IDEAS

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    1. Taylor, A. & Hendrickx, J. & Glineur, F., 2015. "Smooth Strongly Convex Interpolation and Exact Worst-case Performance of First-order Methods," LIDAM Discussion Papers CORE 2015013, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    2. Adrien B. Taylor & Julien M. Hendrickx & François Glineur, 2018. "Exact Worst-Case Convergence Rates of the Proximal Gradient Method for Composite Convex Minimization," Journal of Optimization Theory and Applications, Springer, vol. 178(2), pages 455-476, August.
    3. TAYLOR, Adrien B. & HENDRICKX, Julien M. & François GLINEUR, 2016. "Exact worst-case performance of first-order methods for composite convex optimization," LIDAM Discussion Papers CORE 2016052, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    4. Donghwan Kim & Jeffrey A. Fessler, 2021. "Optimizing the Efficiency of First-Order Methods for Decreasing the Gradient of Smooth Convex Functions," Journal of Optimization Theory and Applications, Springer, vol. 188(1), pages 192-219, January.
    5. Guoyong Gu & Bingsheng He & Xiaoming Yuan, 2014. "Customized proximal point algorithms for linearly constrained convex minimization and saddle-point problems: a unified approach," Computational Optimization and Applications, Springer, vol. 59(1), pages 135-161, October.
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    7. DE KLERK, Etienne & GLINEUR, François & TAYLOR, Adrien B., 2016. "On the Worst-case Complexity of the Gradient Method with Exact Line Search for Smooth Strongly Convex Functions," LIDAM Discussion Papers CORE 2016027, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
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