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High-Order Reduced-Gradient Methods for Composite Variational Inequalities

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  • Nesterov, Yurii

    (Université catholique de Louvain, LIDAM/CORE, Belgium)

Abstract

This paper can be seen as an attempt of rethinking the Extra-Gradient Philosophy for solving Variational Inequality Problems. We show that the properly defined Reduced Gradients can be used instead for finding approximate solutions to Composite Variational Inequalities by the higher-order schemes. Our methods are optimal since their performance is proportional to the lower worst-case complexity bounds for corresponding problem classes. They enjoy the provable hot-start capabilities even being applied to minimization problems. The primal version of our schemes demonstrates a linear rate of convergence under an appropriate uniform monotonicity assumption.

Suggested Citation

  • Nesterov, Yurii, 2024. "High-Order Reduced-Gradient Methods for Composite Variational Inequalities," LIDAM Discussion Papers CORE 2024025, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    • Handle: RePEc:cor:louvco:2024025
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    References listed on IDEAS

    as
    1. Yurii Nesterov, 2018. "Lectures on Convex Optimization," Springer Optimization and Its Applications, Springer, edition 2, number 978-3-319-91578-4, March.
    2. NESTEROV, Yurii, 2013. "Gradient methods for minimizing composite functions," LIDAM Reprints CORE 2510, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    3. Stuart Smith & Leon Lasdon, 1992. "Solving Large Sparse Nonlinear Programs Using GRG," INFORMS Journal on Computing, INFORMS, vol. 4(1), pages 2-15, February.
    4. NESTEROV, Yurii & POLYAK, B.T., 2006. "Cubic regularization of Newton method and its global performance," LIDAM Reprints CORE 1927, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
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