Finite-time and predefined-time dynamical systems for solving strongly pseudomonotone mixed equilibrium problems
TL;DR
This paper establishes the existence and uniqueness of the equilibrium solution, proves the global exponential stability of the associated continuous-time nominal system, and constructs an FT stable PDS whose equilibrium point coincides with the solution of the MEP and derive sufficient conditions for finite-time convergence.
Abstract
In this paper, we propose a class of proximal dynamical systems (PDSs) with finite-time (FT) stability and predefined-time (PdT) stability for solving strongly pseudomonotone mixed equilibrium problems (MEPs) in Hilbert spaces. Unlike existing approaches for variational inequalities and standard equilibrium problems, the proposed framework is developed specifically for MEPs and provides two distinct dynamical models with FT and PdT convergence. Under the assumptions of strong pseudomonotonicity and Lipschitz-type continuity, we establish the existence and uniqueness of the equilibrium solution, prove the global exponential stability of the associated continuous-time nominal system, and show that its discrete-time discretization yields a proximal-type algorithm with linear convergence. Furthermore, we construct an FT stable PDS whose equilibrium point coincides with the solution of the MEP and derive sufficient conditions for finite-time convergence. In addition, we introduce a novel PdT dynamical system that guarantees convergence within a prescribed time independent of the initial conditions. Numerical experiments are provided to demonstrate the effectiveness of the proposed methods, indicating that the PdT system achieves faster convergence compared to the nominal and FT systems. Finally, an application to sparse signal recovery in compressed sensing demonstrates the practical effectiveness of the proposed PdT scheme.