A variant of forward-backward splitting method for the systems of inclusion problem

TL;DR

This paper introduces variants of the forward-backward splitting method for solving systems of inclusion problems, emphasizing computational efficiency and convergence under monotonicity without requiring Lipschitz continuity.

Contribution

It proposes a new algorithm with three variants that reduces computational cost by minimizing operator evaluations and provides convergence analysis under weaker assumptions.

Findings

The algorithm reduces the number of operator evaluations per iteration.

Convergence is established under monotonicity without Lipschitz continuity.

The method demonstrates computational savings compared to Tseng's algorithm.

Abstract

In this paper, we propose variants of forward-backward splitting method for solving the system of splitting inclusion problem. We propose a conceptual algorithm containing three variants, each having a different projection steps. The algorithm consists in two parts, the first and main contains an explicit Armijo-type search in the spirit of the extragradient-like methods for variational inequalities. In the iterative process the operator forward-backward is computed only one time for each inclusion problem, this represent a great computational saving if we compare with Tseng's algorithm, because the computational cost of this operator is very high. The second part of the scheme consists in special projection steps. The convergence analysis of the proposed scheme is given assuming monotonicity on both operators, without assuming Lipschitz continuity on the forward operators.