Primal Methods for Variational Inequality Problems with Functional Constraints
This addresses computational inefficiency in variational inequality problems with multiple constraints, offering a more practical solution for fields like machine learning and operations research, though it is incremental as it builds on existing primal-dual approaches.
The paper tackles functional constrained variational inequality problems by proposing a primal method that avoids needing optimal Lagrange multipliers, achieving complexity matching projection-based methods while using cheaper quadratic programming oracles.
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 requiring any information on the optimal Lagrange multipliers. We establish a non-asymptotic convergence analysis of the algorithm for Minty 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 using significantly cheaper oracles based on quadratic programming. Furthermore, we provide several numerical examples to evaluate the efficacy of our algorithms.