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

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.