A hybrid method for solving systems of operator inclusion problems

TL;DR

This paper introduces a hybrid algorithm combining forward-backward splitting and alternative projection methods to efficiently solve systems of operator inclusion problems with convergence guarantees.

Contribution

It presents a novel hybrid algorithm that reduces computational cost by approximating sets with halfspaces and avoids Lipschitz assumptions for convergence.

Findings

Algorithm converges under monotonicity without Lipschitz continuity.

Significant computational savings by using only one operator evaluation per iteration.

Effective in solving complex systems of operator inclusion problems.

Abstract

In this paper, we propose an algorithm combining the forward-backward splitting method and the alternative projection method for solving the system of splitting inclusion problem. We want to find a point in the interception of a finite number of sets that we don't know, the solution of each component of the system. The algorithm consists of approximate the sets involved in the problem by separates halfspaces which are a known strategy. By finding these halfspaces in each iteration we use only one inclusion problem of the system. The iterations consist of two parts, the first 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 represents a great computational saving because the computational cost of this operator is nothing cheap. The second part consists of special projection step, projecting in the separating halfspace. The convergence analysis of the proposed scheme is given assuming monotonicity all operators, without any Lipschitz continuity assumption.