Averaged proximal reflected gradient method for monotone variational inequalities
Abstract
Projected reflected gradient (PRG) method proposed by Malitsky is efficient for solving monotone variational inequality (MVI), while the existing upper bound of step size is not tight due to the inequality scaling in the theoretical analysis. In this paper, we construct an averaged variant of PRG method for more general MVI and present a novel Lyapunov function to establish convergent theory. This averaged PRG method provides an improvement of the golden ratio algorithm [Y, Malitsky, Math. Program., 184, 383-410, 2020], and the involved step size is compatible with that for the classical methods, such as Popov's extragradient and forward-reflected-backward methods. Moreover, a fully adaptive strategy without linesearch is presented to adjust step sizes, which generates closed-form and potentially much larger step sizes. Numerical experiments on the Nash-Cournot equilibrium, HpHard, and image reconstruction problems demonstrate that the proposed algorithm significantly outperforms existing state-of-the-art methods.
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BibTeX
@misc{chang2026averaged,
title = {{Averaged proximal reflected gradient method for monotone variational inequalities}},
author = {Xiaokai Chang and Jialin Li and Jun Yang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.06409},
eprint = {2609.06409},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.06409},
}
RIS
TY - UNPB
TI - Averaged proximal reflected gradient method for monotone variational inequalities
AU - Xiaokai Chang
AU - Jialin Li
AU - Jun Yang
PY - 2026
DA - 2026/10/15
PB - arXiv
DO - 10.48550/arxiv.2609.06409
UR - https://arxiv.org/abs/2609.06409
AB - Projected reflected gradient (PRG) method proposed by Malitsky is efficient for solving monotone variational inequality (MVI), while the existing upper bound of step size is not tight due to the inequality scaling in the theoretical analysis. In this paper, we construct an averaged variant of PRG method for more general MVI and present a novel Lyapunov function to establish convergent theory. This averaged PRG method provides an improvement of the golden ratio algorithm [Y, Malitsky, Math. Program., 184, 383-410, 2020], and the involved step size is compatible with that for the classical methods, such as Popov's extragradient and forward-reflected-backward methods. Moreover, a fully adaptive strategy without linesearch is presented to adjust step sizes, which generates closed-form and potentially much larger step sizes. Numerical experiments on the Nash-Cournot equilibrium, HpHard, and image reconstruction problems demonstrate that the proposed algorithm significantly outperforms existing state-of-the-art methods.
ER -