In this paper, 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 method, while the second one is a variant of the reflected forward-backward splitting method, which requires only one evaluation of the Lipschitz operator, and one resolvent of the multivalued operator. Under a proper choice of the algorithmic parameters and appropriate conditions on the co-hypomonotone parameter, we theoretically prove that both algorithms achieve $\mathcal{O}(1/k)$ convergence rates on the norm of the residual, where $k$ is the iteration counter. Our results can be viewed as alternatives of a recent class of Halpern-type schemes for root-finding problems.

翻译：本文提出了两种著名的超梯度方法的“涅斯捷罗夫加速”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，其中一个算子为利普希茨连续，另一个可能为多值算子。第一种方案可视为曾氏前向-后向-前向分裂方法的加速变体，而第二种方案是反射前向-后向分裂方法的一种变体，它仅需计算一次利普希茨算子以及一次多值算子的预解式。在算法参数适当选择且共-次单调参数满足恰当条件的前提下，我们从理论上证明了两种算法在残差范数上均能达到$\mathcal{O}(1/k)$的收敛率，其中$k$为迭代计数器。我们的结果可视为近期一类用于求根问题的哈尔彭型方案的可选替代方法。