Constrained variational inequality problems are recognized for their broad applications across various fields including machine learning and operations research. First-order methods have emerged as the standard approach for solving these problems due to their simplicity and scalability. However, they typically rely on projection or linear minimization oracles to navigate the feasible set, which becomes computationally expensive in practical scenarios featuring multiple functional constraints. Existing efforts to tackle such functional constrained variational inequality problems have centered on primal-dual algorithms grounded in the Lagrangian function. These algorithms along with their theoretical analysis often require the existence and prior knowledge of the optimal Lagrange multipliers. In this work, we propose a simple primal method, termed Constrained Gradient Method (CGM), for addressing functional constrained variational inequality problems, without necessitating any information on the optimal Lagrange multipliers. We establish a non-asymptotic convergence analysis of the algorithm for variational inequality problems with monotone operators under smooth constraints. Remarkably, our algorithms match the complexity of projection-based methods in terms of operator queries for both monotone and strongly monotone settings, while utilizing significantly cheaper oracles based on quadratic programming. Furthermore, we provide several numerical examples to evaluate the efficacy of our algorithms.

翻译：约束变分不等式问题因其在机器学习与运筹学等领域的广泛应用而备受关注。一阶方法凭借其简单性和可扩展性，已成为求解此类问题的标准方法。然而，这类方法通常依赖投影或线性最小化预言器来遍历可行集，这在处理含多个函数约束的实际场景中会带来高昂的计算成本。现有针对此类函数约束变分不等式问题的研究主要集中于基于拉格朗日函数的原始-对偶算法。这些算法及其理论分析往往需要最优拉格朗日乘子的存在性及先验知识。本文提出一种简单的原始方法——约束梯度法（CGM），用于求解带函数约束的变分不等式问题，且无需任何关于最优拉格朗日乘子的信息。我们建立了该算法在光滑约束下处理单调算子变分不等式问题的非渐近收敛性分析。值得注意的是，我们的算法在单调和强单调情形下，其算子查询复杂度均与基于投影的方法相当，而实际采用的基于二次规划的预言器成本显著更低。此外，我们通过多个数值算例验证了算法的有效性。