We develop two "Nesterov's accelerated" variants of the well-known extragradient method to approximate a solution of a co-hypomonotone inclusion constituted by the sum of two operators, where one is Lipschitz continuous and the other is possibly multivalued. The first scheme can be viewed as an accelerated variant of Tseng's forward-backward-forward splitting (FBFS) method, while the second one is a Nesterov's accelerated variant of the "past" FBFS scheme, which requires only one evaluation of the Lipschitz operator and one resolvent of the multivalued mapping. Under appropriate conditions on the parameters, we theoretically prove that both algorithms achieve $\mathcal{O}(1/k)$ last-iterate convergence rates on the residual norm, where $k$ is the iteration counter. Our results can be viewed as alternatives of a recent class of Halpern-type methods for root-finding problems. For comparison, we also provide a new convergence analysis of the two recent extra-anchored gradient-type methods for solving co-hypomonotone inclusions.

翻译：我们提出了两类著名的“Nesterov加速”外梯度方法的变体，用于逼近由两个算子之和构成的共次单调包含问题的解，其中一个是Lipschitz连续的，另一个可能是多值的。第一个方案可视为Tseng前向-后向-前向分裂（FBFS）方法的加速变体，而第二个方案则是“过去”FBFS方案的Nesterov加速变体，该方法仅需一次Lipschitz算子求值和一次多值映射的预解计算。在适当的参数条件下，我们从理论上证明两种算法在残差范数上均达到$\mathcal{O}(1/k)$的末次迭代收敛速率，其中$k$为迭代计数器。我们的结果可视为近期一类Halpern型求根方法的替代方案。为便于比较，我们还对最近两种用于求解共次单调包含问题的额外锚定梯度型方法提供了新的收敛性分析。