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On the O(1/t) convergence rate of the projection and contraction methods for variational inequalities with Lipschitz continuous monotone operators

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  • Xingju Cai
  • Guoyong Gu
  • Bingsheng He

Abstract

Nemirovski’s analysis (SIAM J. Optim. 15:229–251, 2005 ) indicates that the extragradient method has the O(1/t) convergence rate for variational inequalities with Lipschitz continuous monotone operators. For the same problems, in the last decades, a class of Fejér monotone projection and contraction methods is developed. Until now, only convergence results are available to these projection and contraction methods, though the numerical experiments indicate that they always outperform the extragradient method. The reason is that the former benefits from the ‘optimal’ step size in the contraction sense. In this paper, we prove the convergence rate under a unified conceptual framework, which includes the projection and contraction methods as special cases and thus perfects the theory of the existing projection and contraction methods. Preliminary numerical results demonstrate that the projection and contraction methods converge twice faster than the extragradient method. Copyright Springer Science+Business Media New York 2014

Suggested Citation

  • Xingju Cai & Guoyong Gu & Bingsheng He, 2014. "On the O(1/t) convergence rate of the projection and contraction methods for variational inequalities with Lipschitz continuous monotone operators," Computational Optimization and Applications, Springer, vol. 57(2), pages 339-363, March.
    • Handle: RePEc:spr:coopap:v:57:y:2014:i:2:p:339-363
    DOI: 10.1007/s10589-013-9599-7
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    References listed on IDEAS

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    1. B. S. He & L. Z. Liao, 2002. "Improvements of Some Projection Methods for Monotone Nonlinear Variational Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 112(1), pages 111-128, January.
    2. Bingsheng He & Li-Zhi Liao & Xiang Wang, 2012. "Proximal-like contraction methods for monotone variational inequalities in a unified framework I: Effective quadruplet and primary methods," Computational Optimization and Applications, Springer, vol. 51(2), pages 649-679, March.
    3. Bingsheng He & Li-Zhi Liao & Xiang Wang, 2012. "Proximal-like contraction methods for monotone variational inequalities in a unified framework II: general methods and numerical experiments," Computational Optimization and Applications, Springer, vol. 51(2), pages 681-708, March.
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    Cited by:

    1. Q. L. Dong & Y. J. Cho & L. L. Zhong & Th. M. Rassias, 2018. "Inertial projection and contraction algorithms for variational inequalities," Journal of Global Optimization, Springer, vol. 70(3), pages 687-704, March.
    2. Xiao-Juan Zhang & Xue-Wu Du & Zhen-Ping Yang & Gui-Hua Lin, 2019. "An Infeasible Stochastic Approximation and Projection Algorithm for Stochastic Variational Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 183(3), pages 1053-1076, 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.
    4. Duong Viet Thong & Phan Tu Vuong & Pham Ky Anh & Le Dung Muu, 2022. "A New Projection-type Method with Nondecreasing Adaptive Step-sizes for Pseudo-monotone Variational Inequalities," Networks and Spatial Economics, Springer, vol. 22(4), pages 803-829, December.
    5. Pham Ngoc Anh, 2023. "New Outer Proximal Methods for Solving Variational Inequality Problems," Journal of Optimization Theory and Applications, Springer, vol. 198(2), pages 479-501, August.
    6. Lateef Olakunle Jolaoso & Maggie Aphane, 2020. "A Generalized Viscosity Inertial Projection and Contraction Method for Pseudomonotone Variational Inequality and Fixed Point Problems," Mathematics, MDPI, vol. 8(11), pages 1-29, November.
    7. Zhong-bao Wang & Xue Chen & Jiang Yi & Zhang-you Chen, 2022. "Inertial projection and contraction algorithms with larger step sizes for solving quasimonotone variational inequalities," Journal of Global Optimization, Springer, vol. 82(3), pages 499-522, March.
    8. Q. L. Dong & Y. C. Tang & Y. J. Cho & Th. M. Rassias, 2018. "“Optimal” choice of the step length of the projection and contraction methods for solving the split feasibility problem," Journal of Global Optimization, Springer, vol. 71(2), pages 341-360, June.
    9. Dang Hieu & Pham Ky Anh & Nguyen Hai Ha, 2021. "Regularization Proximal Method for Monotone Variational Inclusions," Networks and Spatial Economics, Springer, vol. 21(4), pages 905-932, December.

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