KEVIN DOWHON HUANGZhang, ShuzhongShuzhongZhang2025-08-142025-08-142025-07-03https://www.scopus.com/record/display.uri?eid=2-s2.0-105010578149&origin=resultslisthttps://scholars.lib.ntu.edu.tw/handle/123456789/731405In 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 pth-order Lipschitz bound with respect to the original mapping, and the gradient of a (p+1)th-order regularization. The optimal global convergence is guaranteed by including an additional extra-gradient step, while a pth-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.composite operatorsextra-gradient methodhigh-order methodsvariational inequality[SDGs]SDG10journal article10.1137/23M1585258