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

TL;DR

The paper tackles the hardness of solving ε-SVIs in general, focusing on the Minty condition which guarantees an MVI solution. It introduces a novel ExtraGradientEllipsoid algorithm that augments the central-cut ellipsoid with a gradient-descent step to generate strict separating hyperplanes, enabling a polynomial-time method (poly$(d,\, obreak \log(B/ obreak obreak obreak obreak obreak ) , obreak obreak obreak obreak obreak )$) for ε-SVIs under Minty, and it even yields a certificate of MVI infeasibility via strict EVIs when no MVI exists. The approach leverages duality between MVIs and EVIs, and uses a separation oracle framework to handle nonconvex SVIs and low-dimensional MVI solution sets. The results extend to quasar-convex optimization, harmonic games, and two-player Nash or strict CCE computations, while establishing coNP-hardness for deciding Minty feasibility and lower bounds for Minty solving in general. Overall, the work provides the first polynomial-time algorithms under the Minty condition with logarithmic dependence on ε, offering new algorithmic avenues in optimization and game theory and highlighting fundamental complexity limits in related decision problems.

Abstract

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.

Figures (5)

• Figure 1: One step of our ellipsoid algorithm---when the current ellipsoid center ${\bm{a}}^{(t)}$ is not already an $\epsilon$-SVI solution. While $F({\bm{a}}^{(t)})$ separates ${\bm{a}}^{(t)}$ from the set of MVI solutions (in this case a single point), $F(\tilde{{\bm{a}}}^{(t)})$ yields a $\gamma$-strict separating hyperplane, which turns out to be crucial; see \ref{['sec:noopt']}.
• Figure 2: A sequence of $\gamma$-strict separating hyperplanes implies that any ${\bm{x}} \in {\mathcal{K}}$ is far from the boundary of the ellipsoid, assuming that the closest point outside the ellipsoid, labeled ${\bm{x}}'$, belongs to ${\mathcal{X}}$. \ref{['lemma:closeinters']} shows how to make this argument when ${\bm{x}}' \notin {\mathcal{X}}$ by considering instead a point ${\bm{z}} \in {\mathcal{X}}$ that is close to ${\bm{x}}'$.
• Figure 3: The function $f_{\epsilon, \alpha}(x)$, with $\epsilon = 0.1$ and $\alpha = 0$, over the domain $[0,1]$; the y-axis is at a larger scale for the sake of the illustration.
• Figure 4: The vector field in the proof of \ref{['prop:equilcollapse']}.
• Figure 5: The counterexample for the ellipsoid algorithm without extra-gradient as described in \ref{['sec:noopt']}, after $t$ iterations, for $t = 0, 1, \dots, 7, 8$. In each plot, the red polygon is the feasible region where Minty points might still lie, and the blue line/arrow indicates the separating direction $F({\bm{a}}^{(t)})$. The two dashed vertical lines correspond to $x = \pm 1/8$.

Theorems & Definitions (93)

• Definition 1.1: SVIs
• Definition 1.2: MVIs
• Lemma 1.4: Minty's lemma
• Theorem 1.5: Precise version in \ref{['theorem:mainMinty-precise']}
• Lemma 1.6: SVI membership or MVI strict separation
• Theorem 1.7: SVI or strict EVI; precise version in \ref{['theorem:maineither-precise']}
• Definition 1.7: Quasar-convexity
• Corollary 1.8
• Theorem 1.9: Precise version in \ref{['theorem:quasar-precise']}
• Corollary 1.10