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On the Analysis of Misspecified Variational Inequalities with Nonlinear Constraints

Novel Kumar Dey, Mohammad Mahdi Ahmadi, Erfan Yazdandoost Hamedani, Afrooz Jalilzadeh

TL;DR

This work addresses misspecified variational inequalities where both the operator and nonlinear constraints depend on an unknown parameter learned via a secondary VI. It introduces a single-loop augmented Lagrangian algorithm that updates the primal variable, dual multipliers, and the parameter simultaneously, mitigating error propagation from learning. The authors prove bounded dual iterates and an ergodic convergence rate of $\mathcal{O}(1/K)$ for a relaxed Minty VI gap and an infeasibility metric, with numerical experiments showing improvements over decoupled learning-then-optimization approaches. The framework is applicable to nonlinear constraint misspecifications in data-driven VI settings and provides a rigorous, efficient path to convergence in coupled learning-optimization problems.

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.

On the Analysis of Misspecified Variational Inequalities with Nonlinear Constraints

TL;DR

This work addresses misspecified variational inequalities where both the operator and nonlinear constraints depend on an unknown parameter learned via a secondary VI. It introduces a single-loop augmented Lagrangian algorithm that updates the primal variable, dual multipliers, and the parameter simultaneously, mitigating error propagation from learning. The authors prove bounded dual iterates and an ergodic convergence rate of for a relaxed Minty VI gap and an infeasibility metric, with numerical experiments showing improvements over decoupled learning-then-optimization approaches. The framework is applicable to nonlinear constraint misspecifications in data-driven VI settings and provides a rigorous, efficient path to convergence in coupled learning-optimization problems.

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 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.
Paper Structure (16 sections, 6 theorems, 71 equations, 1 figure, 1 table, 1 algorithm)

This paper contains 16 sections, 6 theorems, 71 equations, 1 figure, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Literature Review
  3. Research Gap and Our Contribution
  4. Motivating Example
  5. Cournot with a price-function constraint and unknown slope:
  6. Formulation as a Misspecified Variational Inequality:
  7. Outline of the paper:
  8. Preliminaries and Assumptions
  9. Notation.
  10. Proposed Method
  11. Convergence Analysis
  12. Numerical Experiment
  13. Proof of Lemma \ref{['lemma:lemma2']}
  14. Proof of Lemma \ref{['prop:prop_1']}
  15. Proof of Lemma \ref{['prop:prop2']}
  16. ...and 1 more sections

Key Result

Proposition 2.1

(KKT Conditions) Suppose that a solution pair $(x^*,\theta^*)$ of (eq:PVI) exists and Assumption assum:assum1 holds. Let $f(x, \theta^*) \triangleq [f_1(x, \theta^*), \dots, f_J(x, \theta^*)]^\top$, and the Jacobian matrix $\nabla f(x, \theta^*) \triangleq [\nabla f_1(x, \theta^*), \dots, \nabla f_J

Figures (1)

  • Figure 1: Cournot instances with $(N,D)\in\{(50,5),(50,10),(100,10)\}$.

Theorems & Definitions (17)

  • Definition 2.1
  • Proposition 2.1
  • Lemma 2.2
  • proof
  • Lemma 4.1
  • proof
  • Lemma 4.2: One-step analysis
  • proof
  • Lemma 4.3: Boundedness of Dual Iterates
  • proof
  • ...and 7 more