On the Analysis of Misspecified Variational Inequalities with Nonlinear Constraints

Abstract: In this paper, we study a class of misspecified variational inequalities (VIs) where both the monotone operator and nonlinear convex constraints depend on an unknown parameter learned via a secondary VI. Existing data-driven VI methods typically follow a decoupled learn-then-optimize scheme, causing the approximation error from the learning to propagate the main decision-making problem and hinder convergence. We instead consider a simultaneous approach that jointly solves the main and secondary VIs. To efficiently handle nonlinear constraints with parameter misspecification, we propose a single-loop inexact Augmented Lagrangian method that simultaneously updates the primal decision variables, dual multipliers, and the misspecified parameter. The method combines a forward-reflected-backward step with an Augmented Lagrangian penalty, and explicitly handles misspecification on both the operator and constraint functions. Moreover, we introduce a relaxed performance metric based on the Minty VI gap combined with an aggregated infeasibility metric. By proving boundedness of the dual iterates, we establish $\mathcal{O}(1/K)$ ergodic convergence rates for these metrics. Numerical Experiments are provided to showcase the superior performance of our algorithm compared to state-of-the-art methods.