Expected Variational Inequalities

2025-02-25

AI Summary

This paper addresses the computational intractability of solving variational inequalities (VIs), particularly in non-monotone and non-convex settings. To overcome this fundamental limitation, we propose the Expectation Variational Inequality (EVI) paradigm: a novel relaxation of classical pointwise VI constraints to distribution-level expectation constraints. Methodologically, we formally define the EVI framework, integrating distributional optimization, expectation-constrained modeling, and polynomial-time algorithm design. Theoretical contributions are twofold: (1) We establish that EVI remains solvable in polynomial time even for general non-monotone operators—circumventing the inherent computational hardness of standard VIs; (2) We provide a unified characterization of correlated equilibria and several classes of computationally challenging games—including smooth games, games with coupled constraints, and games with non-concave utilities—thereby strictly generalizing existing models.

Technology Category

• Game Theory and Economic Paradigms: Equilibrium
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Multiagent Systems: Mechanism Design

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution

Abstract

Variational inequalities (VIs) encompass many fundamental problems in diverse areas ranging from engineering to economics and machine learning. However, their considerable expressivity comes at the cost of computational intractability. In this paper, we introduce and analyze a natural relaxation -- which we refer to as expected variational inequalities (EVIs) -- where the goal is to find a distribution that satisfies the VI constraint in expectation. By adapting recent techniques from game theory, we show that, unlike VIs, EVIs can be solved in polynomial time under general (nonmonotone) operators. EVIs capture the seminal notion of correlated equilibria, but enjoy a greater reach beyond games. We also employ our framework to capture and generalize several existing disparate results, including from settings such as smooth games, and games with coupled constraints or nonconcave utilities.

Problem

• Address computational intractability in variational inequalities.
• Introduce expected variational inequalities for efficient solutions.
• Generalize existing results in game theory and beyond.

Innovation

• Expected Variational Inequalities introduced
• Polynomial time solvable under nonmonotone operators
• Generalizes existing results from game theory