发表机构
Tiangong University; University of Hertfordshire; Sefako Makgatho Health Sciences University; Zhejiang Normal University; China Medical University; Academy of Romanian Scientists（天津工业大学; 赫特福德大学; 塞法科·马卡托健康科学大学; 浙江师范大学; 中国医科大学; 罗马尼亚科学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出约束变分不等式的原始方法理论，给出最优收敛速率、可行性权衡及下界，并设计OPCGM等算法达到最优复杂度。
AI 中文摘要
自Zhang等人（Zhang2025）首次提出针对受凸函数约束的单调变分不等式的纯原始方法以来，该领域一直缺乏对此类预言机复杂度的完整理解。我们发展了一套理论，解决了其开创性框架留下的关键空白。我们提出的最优原始约束梯度法（OPCGM）建立了限制在局部线性约束近似下的算法的收敛速率和下界。对于强单调算子，我们证明了一种改进的原始梯度法在最优性间隙和约束违反两方面均达到最优的O(1/T)速率，消除了先前工作中存在的次优指数。对于Lipschitz单调算子，我们提出了一种原始外梯度变体，仅使用二次规划预言机即可达到最优的O(1/ε)间隙复杂度，代价是平均迭代具有恒定的渐近可行性违反；我们证明了在具有正曲率的光滑凸约束上，对于自然类别的常数步长原始外梯度方法，第一步必然不可行。我们建立了限制在局部线性近似下的原始方法的下界复杂度，证明对于Lipschitz常数按Θ(1/ε)缩放的Lipschitz单调变分不等式，复杂度为Ω(1/ε^2)；对于标准Lipschitz单调变分不等式，复杂度为Ω(1/ε)。我们设计了一种单循环原始方法，仅需域半径估计即可达到O(1/√T)的间隙速率，代价是问题依赖的渐近可行性常数，我们证明该常数是不可避免的。对于强单调问题，我们证明了最后迭代以最优的O(1/T)速率收敛。
英文摘要
Since Zhang et al. \cite{Zhang2025} introduced the first purely primal methods for monotone variational inequalities subject to convex functional constraints, the field has lacked a complete understanding of the complexity limits of this oracle class. We develop a theory that resolves the critical gaps left by their pioneering framework. Our Optimal Primal Constrained Gradient Method (OPCGM) establishes convergence rates and lower bounds for algorithms restricted to local linear constraint approximations. For strongly monotone operators, we prove that a refined primal gradient method achieves the optimal $\mathcal{O}(1/T)$ rate for both optimality gap and constraint violation, eliminating the suboptimal exponent present in prior work. For Lipschitz monotone operators, we propose a primal extragradient variant that achieves the optimal $\mathcal{O}(1/ε)$ gap complexity using only quadratic programming oracles, at the cost of a constant asymptotic feasibility violation for the averaged iterate; we prove that the first half-step is necessarily infeasible for the natural class of constant-stepsize primal extragradient methods on smooth convex constraints with positive curvature. We establish lower complexity bounds for primal methods restricted to local linear approximations, proving $Ω(1/ε^2)$ for Lipschitz monotone variational inequalities with Lipschitz constant scaling as $Θ(1/ε)$, and $Ω(1/ε)$ for standard Lipschitz monotone variational inequalities. We design a single-loop primal method that achieves an $\mathcal{O}(1/\sqrt{T})$ gap rate with only a domain-radius estimate, at the cost of a problem-dependent asymptotic feasibility constant that we prove is unavoidable. For strongly monotone problems, we prove that the last iterate converges at the optimal $\mathcal{O}(1/T)$ rate.
