We study monotone inclusions and monotone variational inequalities, as well as their generalizations to non-monotone settings. We first show that the Extra Anchored Gradient (EAG) algorithm, originally proposed by Yoon and Ryu [2021] for unconstrained convex-concave min-max optimization, can be applied to solve the more general problem of Lipschitz monotone inclusion. More specifically, we prove that the EAG solves Lipschitz monotone inclusion problems with an \emph{accelerated convergence rate} of $O(\frac{1}{T})$, which is \emph{optimal among all first-order methods} [Diakonikolas, 2020, Yoon and Ryu, 2021]. Our second result is a new algorithm, called Extra Anchored Gradient Plus (EAG+), which not only achieves the accelerated $O(\frac{1}{T})$ convergence rate for all monotone inclusion problems, but also exhibits the same accelerated rate for a family of general (non-monotone) inclusion problems that concern negative comonotone operators. As a special case of our second result, EAG+ enjoys the $O(\frac{1}{T})$ convergence rate for solving a non-trivial class of nonconvex-nonconcave min-max optimization problems. Our analyses are based on simple potential function arguments, which might be useful for analysing other accelerated algorithms.

翻译：我们研究单调包容和单调差异性,以及它们对于非单调环境的概括化。 我们首先显示, 由Yoon 和 Luno [2021年] 提出的用于不受约束的 convex 康调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调, 可以用于解决Lipschitz 单调调调调调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调和调