A Polynomial-Time Algorithm for Variational Inequalities under the Minty Condition
It solves a foundational optimization problem with broad implications in game theory and nonsmooth optimization, representing a major theoretical advance.
The paper presents the first polynomial-time algorithm for solving variational inequalities under the Minty condition, achieving complexity polynomial in dimension and log(1/ε), and applies it to problems like minimizing quasar-convex functions and computing Nash equilibria in harmonic games.
Solving variational inequalities (SVIs) is a foundational problem at the heart of optimization. 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 admits a solution. In this paper, we establish the first polynomial-time algorithm -- that is, with complexity growing polynomially in the dimension $d$ and $\log(1/ε)$ -- for solving $ε$-SVIs for Lipschitz continuous mappings under the Minty condition. Prior approaches either incurred an exponentially worse dependence on $1/ε$ or made restrictive assumptions. To do so, we introduce a new variant of the ellipsoid algorithm whereby 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, thereby establishing that the disjunction of those two problems is polynomial-time solvable even though each problem is individually intractable. We provide several extensions and new applications of our main results. Most notably, we obtain the first polynomial-time algorithms for i) globally minimizing a (potentially nonsmooth) quasar-convex function, and ii) computing Nash equilibria in multi-player harmonic games. Finally, in two-player general-sum concave games, we give the first polynomial-time algorithm that outputs either a Nash equilibrium or a strict coarse correlated equilibrium.