Forward–Partial Inverse–Forward Splitting for Solving Monotone Inclusions

In this paper, we provide a splitting method for finding a zero of the sum of a maximally monotone operator, a Lipschitzian monotone operator, and a normal cone to a closed vector subspace of a real Hilbert space. The problem is characterised by a simpler monotone inclusion involving only two operators: the partial inverse of the maximally monotone operator with respect to the vector subspace and a suitable Lipschitzian monotone operator. By applying the Tseng’s method in this context, we obtain a fully split algorithm that exploits the whole structure of the original problem and generalises partial inverse and Tseng’s methods. Connections with other methods available in the literature are provided, and the flexibility of our setting is illustrated via applications to some inclusions involving m\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m$$\end{document} maximally monotone operators, to primal-dual composite monotone inclusions, and to zero-sum games.