发表机构
Zhejiang Normal University（浙江师范大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
提出一种无参数自适应反射梯度方法求解单调变分不等式，无需Lipschitz常数和额外评估，证明无条件弱收敛及强单调下的R-线性收敛，并实现Ω(L)加速。
AI 中文摘要
我们分析了一种用于单调 L-Lipschitz 映射 B 的 VI(C,B) 问题的每次迭代仅需一次函数评估的方法：x_{k+1}=P_C(x_k-λB(u_k))，其中 u_k=x_k+θ_k(x_k-x_{k-1})+β_k(x_k-u_{k-1})，且 θ_k+β_k=1。对于常数步长和可求和滤波器（∑_k β_k<∞），我们通过具有精确有理耗散预算的 Lyapunov 函数证明了弱收敛性，其中包括 Malitsky 的反射梯度方法。该方法对可求和算子误差具有鲁棒性；若 C 有界，则常数步长范围可达 λ<(√2-1)/L。对于无界 C，这归结为先验有界性陈述，通过耗散交易认证到 λL=0.387；对于多面体 C 上的仿射 B，有 dist(x_n,S)→0 且 ∑_n dist^2(x_n,S)<∞，对于有界 C 有强收敛性，以及在面识别后的 R-线性收敛速率。主要结果是一种受保护的自适应步长规则，无需 L 且无需额外评估，被证明无条件弱收敛：数据驱动的 Lyapunov 权重从耗散预算中消除了 L。对于 C=H 上的仿射 B，通过旋转下界证明了 λL=1/√3 的锐度；该常数对于无约束非线性 B 也是锐利的，且算子值平方可和时收敛。低于 1/(√3 L) 的无条件收敛归结为边际极点绝对稳定性陈述；投影情形仍然开放。在强单调性下，我们证明了具有显式收缩因子的 R-线性收敛，并认证了相对于常数步长的 Ω(L) 加速。数值实验证实了这些增益。
英文摘要
We analyze a one-evaluation-per-iteration method for $\mathrm{VI}(C,B)$ with monotone $L$-Lipschitz $B$: $x_{k+1}=P_C(x_k-λB(u_k))$, $u_k=x_k+θ_k(x_k-x_{k-1})+β_k(x_k-u_{k-1})$, $θ_k+β_k=1$. For constant step and summable filter ($\sum_kβ_k<\infty$) we prove weak convergence via a Lyapunov function with exact rational dissipation budgets, including Malitsky's reflected gradient method. It is robust to summable operator errors; if $C$ is bounded, the constant-step range reaches $λ<(\sqrt2-1)/L$. For unbounded $C$ this reduces to an a priori boundedness statement, certified to $λL=0.387$ by dissipation trading; for affine $B$ on polyhedral $C$, $\dist(x_n,S)\to0$ with $\sum_n\dist^2(x_n,S)<\infty$, strong convergence for bounded $C$, and $R$-linear rates after face identification. The main result is a safeguarded adaptive step-size rule needing no $L$ and no extra evaluations, proved weakly convergent unconditionally: a data-driven Lyapunov weight removes $L$ from the dissipation budgets. For affine $B$ with $C=\mathcal H$, sharpness of $λL=1/\sqrt3$ via a rotation lower bound; the same constant is sharp for unconstrained nonlinear $B$, with convergence for square-summable operator values. Unconditional convergence below $1/(\sqrt3\,L)$ reduces to a marginal-pole absolute-stability statement; the projected case remains open. Under strong monotonicity we prove $R$-linear convergence with explicit contraction, and certify $Ω(L)$ speedups over constant steps. Numerics confirm the gains.
Comments44 pages, 3 figures
