A fixed-time stable forward-backward dynamical system for solving generalized monotone inclusions

TL;DR

This paper introduces a new fixed-time stable forward-backward dynamical system for solving generalized monotone inclusion problems in Hilbert spaces, with convergence analysis and applications to various optimization problems.

Contribution

It develops a novel fixed-time stable dynamical system under weaker monotonicity conditions and provides a discretized algorithm with convergence guarantees.

Findings

Proves fixed-time stability of the proposed system.

Establishes convergence of the discretized algorithm.

Demonstrates effectiveness through numerical examples.

Abstract

We propose a forward-backward splitting dynamical system for solving inclusion problems of the form $0\in A(x)+B(x)$ in Hilbert spaces, where $A$ is a maximal operator and $B$ is a single-valued operator. Involved operators are assumed to satisfy a generalized monotonicity condition, which is weaker than the standard monotone assumptions. Under mild conditions on parameters, we establish the fixed-time stability of the proposed dynamical system. In addition, we consider an explicit forward Euler discretization of the dynamical system leading to a new forward backward algorithm for which we present the convergence analysis. Applications to other optimization problems such as Constrained Optimization Problems (COPs), Mixed Variational Inequalities (MVIs), and Variational Inequalities (VIs) are presented and some numerical examples are given to illustrate the theoretical results.