Forward-backward-half forward splitting algorithm with deviations

In this paper, we present a forward-backward-half forward splitting algorithm with deviations for solving the structured monotone inclusion problem composed of a maximally monotone operator, a maximally monotone and Lipschitz continuous operator and a cocoercive operator. The weak and linear convergence of the proposed algorithms is established. The aim of introducing deviations is to improve the performance of the forward-backward-half forward splitting algorithm through suitable choices of the deviations. A numerical example is given to show that the two-step inertial variant of the forward-backward-half forward splitting algorithms with deviations outperforms the existing methods in Briceño-Arias and Davis Forward-backward-half forward algorithm for solving monotone inclusions, SIAM J Optimiz, 2017;28(4):2839–2871. doi: 10.1137/17M1120099, Zong et al. An accelerated forward-backward-half forward splitting algorithm for monotone inclusion with applications to image restoration, Optimization, 2022;73(2):401–428. doi: 10.1080/02331934.2022.2107926 and Fan et al. Convergence of an inertial shadow Douglas–Rachford splitting algorithm for monotone inclusions, Numer Func Anal Optim, 2021;42(14):1627–1644. doi: 10.1080/01630563.2021.2001749.