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High-Order Reduced-Gradient Methods for Composite Variational Inequalities

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arxiv 2311.15154 v2 pith:7S5LC5G6 submitted 2023-11-26 math.OC

Yurii Nesterov This is my paper · ORCID
classification math.OC
keywords variationalcompositeinequalitiesmethodsproblemsschemesappliedappropriate
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This paper can be seen as an attempt of rethinking the {\em Extra-Gradient Philosophy} for solving Variational Inequality Problems. We show that the properly defined {\em 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.

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Cited by 3 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Convergence Analysis of the Restarted Moving-Anchored Extra-Gradient Method in the Absence of Local Lipschitz Continuity

    math.OC 2026-07 accept novelty 7.0 of 10

    The MAEG-R method achieves convergence for monotone inclusions with merely continuous operators via a moving-anchor restart strategy, while preserving O(1/k) complexity in the Lipschitz case.

  2. Universal Reduced-Operator Method and High-Order Global Curvature Bounds

    math.OC 2025-11 conditional novelty 7.0 of 10

    A universal high-order method for variational inequalities whose complexity is governed by a new Global Curvature Bound of the operator, matching optimal rates under Hölder smoothness.

  3. Solving Convex-Concave Problems with $\tilde{\mathcal{O}}(\epsilon^{-4/7})$ Second-Order Oracle Complexity

    math.OC 2025-06 conditional novelty 7.0 of 10

    A new triple-loop algorithm, Minimax-AIPE, solves convex-concave minimax problems with tilde O(epsilon^{-4/7}) second-order oracle calls, improving the previous O(epsilon^{-2/3}).

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