Extragradient-Type Methods with $\mathcal{O}(1/k)$ Convergence Rates for Co-Hypomonotone Inclusions

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.