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An Approximation-Based Regularized Extra-Gradient Method for Monotone Variational Inequalities.

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Titel: An Approximation-Based Regularized Extra-Gradient Method for Monotone Variational Inequalities.
Autoren: Huang, Kevin1 (AUTHOR) kdhuang@ntu.edu.tw, Zhang, Shuzhong2 (AUTHOR) zhangs@umn.edu
Quelle: SIAM Journal on Optimization. 2025, Vol. 35 Issue 3, p1469-1497. 29p.
Schlagwörter: VARIATIONAL inequalities (Mathematics), APPROXIMATION algorithms, LIPSCHITZ continuity, OPTIMIZATION algorithms, SUBGRADIENT methods
Abstract: In this paper, we propose a general extra-gradient scheme for solving monotone variational inequalities (VI), referred to here as the Approximation-based Regularized Extra-gradient method (ARE). The first step of ARE solves a VI subproblem, where the associated operator consists of an approximation operator satisfying a \(p{\textrm {th}}\) -order Lipschitz bound with respect to the original mapping, and the gradient of a \((p+1){\textrm {th}}\) -order regularization. The optimal global convergence is guaranteed by including an additional extra-gradient step, while a \(p{\textrm {th}}\) -order superlinear local convergence is shown to hold if the VI is strongly monotone. The proposed ARE is a broad scheme, in the sense that a variety of solution methods can be formulated within this framework as different manifestations of approximations, and their iteration complexities would follow through in a unified fashion. The ARE framework relates to the first-order methods, while opening up possibilities to developing higher-order methods specifically for structured problems that guarantee the optimal iteration complexity bounds. [ABSTRACT FROM AUTHOR]
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Abstract:In this paper, we propose a general extra-gradient scheme for solving monotone variational inequalities (VI), referred to here as the Approximation-based Regularized Extra-gradient method (ARE). The first step of ARE solves a VI subproblem, where the associated operator consists of an approximation operator satisfying a \(p{\textrm {th}}\) -order Lipschitz bound with respect to the original mapping, and the gradient of a \((p+1){\textrm {th}}\) -order regularization. The optimal global convergence is guaranteed by including an additional extra-gradient step, while a \(p{\textrm {th}}\) -order superlinear local convergence is shown to hold if the VI is strongly monotone. The proposed ARE is a broad scheme, in the sense that a variety of solution methods can be formulated within this framework as different manifestations of approximations, and their iteration complexities would follow through in a unified fashion. The ARE framework relates to the first-order methods, while opening up possibilities to developing higher-order methods specifically for structured problems that guarantee the optimal iteration complexity bounds. [ABSTRACT FROM AUTHOR]
ISSN:10526234
DOI:10.1137/23M1585258