Solving (Stampacchia) 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 -- 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 $1/ε$ (and other natural parameters of the problem) or made more restrictive assumptions, such as monotonicity. To do so, we introduce a new variant of the ellipsoid algorithm whereby separating hyperplanes are obtained after taking a 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 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.

翻译：求解（Stampacchia）变分不等式是优化领域的核心基础问题。然而，这种表达力以计算难度为代价。因此，大部分研究聚焦于刻画能够规避此类难解性壁垒的特定子类。可追溯至20世纪60年代的一个经典性质是明提条件，该条件假设明提变分不等式存在解。本文针对满足明提条件的Lipschitz连续映射，建立了首个多项式时间算法（其复杂度随维度$d$和$\log(1/ε)$呈多项式增长）用于求解$ε$-SVI。先前的方法要么对$1/ε$（及问题的其他自然参数）呈指数级依赖，要么施加了更严格的假设（如单调性）。为此，我们提出了一种椭球算法的新变体，其中分割超平面通过在椭球中心执行下降步骤后获得。尽管SVI可能非凸且非满维，该算法仍能成功运作。此外，当算法应用于无MVI解的实例且无法识别SVI解时，会生成一个关于MVI不可行性的简洁证。我们还证明了判定明提条件是否成立是$\mathsf{coNP}$完全的，从而表明尽管这两个问题各自不可解，但其析取可在多项式时间内求解。我们给出了主要结果的若干扩展与新应用。最值得注意的是，我们在多人调和博弈中首次获得了计算纳什均衡的多项式时间算法。最后，在两人一般和凹博弈中，我们给出了首个能够输出纳什均衡或严格粗相关均衡的多项式时间算法。