Forward--partial inverse--forward splitting for solving monotone inclusions

TL;DR

This paper introduces a novel splitting algorithm for solving complex monotone inclusions in Hilbert spaces, combining partial inverse and Tseng's methods to handle multiple operators efficiently.

Contribution

The paper develops a new splitting method that generalizes existing approaches by exploiting the structure of the problem involving multiple monotone operators.

Findings

The algorithm effectively solves inclusions with multiple maximally monotone operators.

Connections established with existing methods enhance understanding of the algorithm's scope.

Applications demonstrated in zero-sum games and primal-dual problems.

Abstract

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 vectorial subspace of a real Hilbert space. The problem is characterized by a simpler monotone inclusion involving only two operators: the partial inverse of the maximally monotone operator with respect to the vectorial subspace and a suitable lipschitzian monotone operator. By applying the Tseng's method in this context we obtain a splitting algorithm that exploits the whole structure of the original problem and generalizes partial inverse and Tseng's methods. Connections with other methods available in the literature and applications to inclusions involving $m$ maximally monotone operators, to primal-dual composite monotone inclusions, and to zero-sum games are provided.