Novel Dynamical Systems with Finite-Time and Fixed-Time Stability for Generalized Inverse Mixed Variational Inequality Problems
Nam Van Tran
TL;DR
The paper tackles the generalized inverse mixed variational inequality problem (GIMVIP) and develops two first-order continuous-time dynamical systems that achieve finite-time and fixed-time convergence to the GIMVIP solution. It leverages Lyapunov-based analysis and operator-theoretic assumptions (monotonicity, cocoercivity, and Lipschitz properties) to establish convergence guarantees, and systematically derives discretized proximal-point–type algorithms via forward Euler discretization that retain fixed-time convergence. The work provides explicit time bounds and conditions under which convergence is guaranteed, along with a discretization framework supported by a convergence analysis inspired by existing fixed-time theory. Numerical experiments in a simple setting validate the fast convergence and illustrate the practical viability of the proposed dynamics and their discretizations for inverse equilibrium problems.
Abstract
This paper investigates a class of generalized inverse mixed variational inequality problems (GIMVIPs), which consist in finding a vector $\overline{w}\in \R^d$ such that \[ F(\bar w)\in Ω\quad \text{and} \quad \langle h(\bar w), v-F(\bar w) \rangle + g(v)-g(F(\bar w)) \ge 0, \quad \forall v\in Ω, \] where \(h,F:\R^d\to\R^d\) are single-valued operators, \(g:Ω\to\R\cup\{+\infty\}\) is a proper function, and \(Ω\) is a closed convex set. Two novel continuous-time dynamical systems are proposed to analyze the finite-time and fixed-time stability of solutions to GIMVIPs in finite-dimensional Hilbert spaces. Under suitable assumptions on the operators and model parameters, Lyapunov-based techniques are employed to establish finite-time and fixed-time convergence of the generated trajectories. While both systems exhibit accelerated convergence, the settling time of the finite-time stable system depends on the initial condition, whereas the fixed-time stable system admits a uniform upper bound on the convergence time that is independent of the initial state. Moreover, an explicit forward Euler discretization of the continuous-time dynamics leads to a proximal point-type algorithm that preserves the fixed-time convergence property. Rigorous convergence analysis of the resulting iterative scheme is provided. A numerical experiment is presented to demonstrate the effectiveness of the proposed methods.