First-order Methods for Variational Inequality Problems with Semi-Infinite Constraints

Abstract

This paper studies first-order methods for monotone variational inequalities with semi-infinite constraints and presents this model as a framework covering constrained saddle-point problems, robust optimization, distributionally robust function-constrained optimization, and generalized Nash equilibrium formulations. Since feasibility must hold over infinitely many indexed constraints, direct projection onto the full feasible region is generally difficult. Therefore, we propose a single-loop primal-dual constraint extrapolation method (PD-ConEx) that updates primal variables, Lagrange multipliers, and worst-case constraint indices using first-order information, linearization, extrapolation, and projections only onto simple sets. We establish convergence guarantees for smooth and nonsmooth regimes, showing how the achievable rates depend on strong monotonicity and strong concavity of the index block. For coupled convex-concave saddle-point problems, strong convexity in the minimization block can accelerate its smooth terms even without strong concavity in the maximization block. We also develop an adaptive smooth variant with $O(1/K)$ VI-gap and feasibility guarantees without an a priori bound on generated multipliers. Moment-based and Wasserstein distributionally robust models fit within the framework, and numerical experiments on robust quadratic instances and distributionally robust learning illustrate the method's optimality and feasibility behavior.