A Polynomial-Time Algorithm for Variational Inequalities under the Minty Condition

2025-04-04

AI Summary

This paper addresses the ε-accurate Stampacchia variational inequality (SVI) problem for Lipschitz continuous mappings satisfying the Minty condition, presenting the first polynomial-time algorithm—overcoming prior reliance on exponential precision or strong monotonicity assumptions. Methodologically, it introduces a novel variant of the ellipsoid method: separation hyperplanes are constructed via gradient descent, integrated with duality theory for variational inequalities and computational complexity analysis. Key contributions include: (1) the first polynomial-time solver for SVI; (2) a succinct certificate for Minty infeasibility; (3) a proof that deciding the Minty condition is coNP-complete; and (4) polynomial-time algorithms—derived concurrently—for monotone variational inequalities, global minimization of nonsmooth quasiconvex functions, and computing Nash equilibria in multi-player harmonic games.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Non-convex Optimization
• Game Theory and Economic Paradigms: 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
• Security and Privacy: Large-scale security measurements

Abstract

Solving (Stampacchia) variational inequalities (SVIs) is a foundational problem at the heart of optimization, with a host of critical applications ranging from engineering to economics. However, this expressivity comes at the cost of computational hardness. As a result, most research has focused on carving out specific subclasses that elude those intractability barriers. A classical property that goes back to the 1960s is the Minty condition, which postulates that the Minty VI (MVI) problem -- the weak dual of the SVI problem -- admits a solution. In this paper, we establish the first polynomial-time algorithm -- that is, with complexity growing polynomially in the dimension $d$ and $log(1/epsilon)$ -- for solving $epsilon$-SVIs for Lipschitz continuous mappings under the Minty condition. Prior approaches either incurred an exponentially worse dependence on $1/epsilon$ (and other natural parameters of the problem) or made overly restrictive assumptions -- such as strong monotonicity. To do so, we introduce a new variant of the ellipsoid algorithm wherein separating hyperplanes are obtained after taking a gradient descent step from the center of the ellipsoid. It succeeds even though the set of SVIs can be nonconvex and not fully dimensional. Moreover, when our algorithm is applied to an instance with no MVI solution and fails to identify an SVI solution, it produces a succinct certificate of MVI infeasibility. We also show that deciding whether the Minty condition holds is $mathsf{coNP}$-complete. We provide several extensions and new applications of our main results. Specifically, we obtain the first polynomial-time algorithms for i) solving monotone VIs, ii) globally minimizing a (potentially nonsmooth) quasar-convex function, and iii) computing Nash equilibria in multi-player harmonic games.

Problem

• Solving polynomial-time variational inequalities under Minty condition
• Developing efficient algorithm for Lipschitz continuous mappings
• Extending results to monotone VIs and quasar-convex optimization

Innovation

• Polynomial-time algorithm for Minty condition VIs
• New ellipsoid variant with gradient descent
• Certifies infeasibility when no solution exists