发表机构
School of Science, Lanzhou University of Technology; School of Mathematics and Statistics, Xianyang Normal University（兰州理工大学理学院; 咸阳师范学院数学与统计学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出平均近端反射梯度方法，通过新Lyapunov函数建立收敛理论，改进黄金比例算法，采用自适应步长，在多个问题上优于现有方法。
AI 中文摘要
Malitsky提出的投影反射梯度（PRG）方法在求解单调变分不等式（MVI）时是高效的，然而由于理论分析中的不等式缩放，现有的步长上界并不紧致。在本文中，我们针对更一般的MVI构建了PRG方法的平均变体，并提出了一种新颖的Lyapunov函数来建立收敛理论。这种平均PRG方法改进了黄金比例算法[Y. Malitsky, Math. Program., 184, 383-410, 2020]，并且所涉及的步长与经典方法（如Popov的外梯度和前向-反射-后向方法）中的步长相容。此外，我们提出了一种无需线搜索的完全自适应策略来调整步长，该策略生成闭式且可能大得多的步长。在Nash-Cournot均衡、HpHard和图像重建问题上的数值实验表明，所提出的算法显著优于现有的最先进方法。
英文摘要
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.
Journal refpublished in Computational Optimization and Applications, 2026
