Forward-Primal-Dual-Half-Forward Algorithm for Splitting Four Operators

In this article, we propose a splitting algorithm to find zeros of the sum of four maximally monotone operators in real Hilbert spaces. In particular, we consider a Lipschitz operator, a cocoercive operator, and a linear composite term. In the case where the Lipschitz operator is absent, our method reduces to the Condat-Vũ algorithm. On the other hand, when the linear composite term is absent, the algorithm reduces to the Forward-Backward-Half-Forward algorithm (FBHF). Additionally, in each case, the set of step-sizes that guarantee the weak convergence of those methods are recovered. Therefore, our algorithm can be seen as a combination of Condat-Vũ and FBHF. Moreover, we propose extensions and applications of our method in multivariate monotone inclusions and saddle point problems. Finally, we present a numerical experiment on image deblurring problems.