This paper settles an open and challenging question pertaining to the design of simple and optimal high-order methods for solving smooth and monotone variational inequalities (VIs). A VI involves finding $x^\star \in \mathcal{X}$ such that $\langle F(x), x - x^\star\rangle \geq 0$ for all $x \in \mathcal{X}$. We consider the setting in which $F$ is smooth with up to $(p-1)^{th}$-order derivatives. For $p = 2$, the cubic regularized Newton method was extended to VIs with a global rate of $O(\epsilon^{-1})$. An improved rate of $O(\epsilon^{-2/3}\log\log(1/\epsilon))$ can be obtained via an alternative second-order method, but this method requires a nontrivial line-search procedure as an inner loop. Similarly, high-order methods based on line-search procedures have been shown to achieve a rate of $O(\epsilon^{-2/(p+1)}\log\log(1/\epsilon))$. As emphasized by Nesterov, however, such procedures do not necessarily imply practical applicability in large-scale applications, and it would be desirable to complement these results with a simple high-order VI method that retains the optimality of the more complex methods. We propose a $p^{th}$-order method that does \textit{not} require any line search procedure and provably converges to a weak solution at a rate of $O(\epsilon^{-2/(p+1)})$. We prove that our $p^{th}$-order method is optimal in the monotone setting by establishing a matching lower bound under a generalized linear span assumption. Our method with restarting attains a linear rate for smooth and strictly monotone VIs and a local superlinear rate for smooth and strongly monotone VIs. Our method also achieves a global rate of $O(\epsilon^{-2/p})$ for solving smooth and nonmonotone VIs satisfying the Minty condition and when augmented with restarting it attains a global linear and local superlinear rate for smooth and nonmonotone VIs satisfying the strictly/strong Minty condition.

翻译：摘要：本文解决了一个关于设计用于求解光滑单调变分不等式的简单且最优高阶方法的开放且具有挑战性的问题。变分不等式涉及寻找 $x^\star \in \mathcal{X}$，使得对于所有 $x \in \mathcal{X}$ 有 $\langle F(x), x - x^\star\rangle \geq 0$。我们考虑 $F$ 具有高达 $(p-1)$ 阶导数的光滑性设定。对于 $p = 2$，三次正则化牛顿方法被推广到变分不等式，达到了 $O(\epsilon^{-1})$ 的全局速率。通过一种替代的二阶方法可以获得改进的 $O(\epsilon^{-2/3}\log\log(1/\epsilon))$ 速率，但该方法需要内部循环中执行非平凡的线搜索过程。类似地，基于线搜索过程的高阶方法已被证明能达到 $O(\epsilon^{-2/(p+1)}\log\log(1/\epsilon))$ 的速率。然而，正如Nesterov所强调的，这些过程不一定意味着在大规模应用中的实际适用性，因此有必要用保留更复杂方法最优性的简单高阶变分不等式方法来补充这些结果。我们提出了一种 $p$ 阶方法，该方法不需要任何线搜索过程，并且被证明以 $O(\epsilon^{-2/(p+1)})$ 的速率收敛到弱解。通过建立广义线性跨度假设下的匹配下界，我们证明了所提出的 $p$ 阶方法在单调设定下是最优的。我们的方法通过重启对于光滑且严格单调的变分不等式达到线性速率，对于光滑且强单调的变分不等式达到局部超线性速率。该方法在求解满足Minty条件的光滑非单调变分不等式时，也实现了 $O(\epsilon^{-2/p})$ 的全局速率，并且当配合重启时，对于满足严格/强Minty条件的光滑非单调变分不等式，能达到全局线性和局部超线性速率。