Novel Dynamical Systems with Finite-Time and Predefined-Time Stability for Generalized Inverse Mixed Variational Inequality Problems

TL;DR

This paper introduces two new continuous-time dynamical systems with finite-time and predefined-time stability for solving generalized inverse mixed variational inequality problems, providing explicit convergence guarantees and practical algorithms.

Contribution

The paper proposes novel dynamical systems with finite-time and predefined-time stability for GIMVIPs, including explicit convergence analysis and a discretized iterative algorithm.

Findings

Both systems achieve accelerated convergence.

Predefined-time system has a fixed upper bound on convergence time.

Numerical experiments confirm the effectiveness of the methods.

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 \[ F(\bar w)\in \Omega \quad \text{and} \quad \langle h(\bar w), v-F(\bar w) \rangle + g(v)-g(F(\bar w)) \ge 0, \quad \forall v\in \Omega, \] where \(h,F:\R^d\to\R^d\) are single-valued operators, \(g:\Omega\to\R\cup\{+\infty\}\) is a proper function, and \(\Omega\) is a closed convex set. Two novel continuous-time dynamical systems are proposed to study the finite-time and predefined-time stability of solutions to GIMVIPs in finite-dimensional Hilbert spaces. Under suitable assumptions on the involved operators and model parameters, Lyapunov-based techniques are employed to establish finite-time and predefined-time convergence of the generated trajectories. Although both dynamical systems exhibit accelerated convergence, the settling time of the finite-time stable system depends on the initial condition, whereas the predefined-time stable system admits a uniform upper bound on the convergence time that is independent of the initial state and can be explicitly prescribed through user-selected parameters. Moreover, by applying a forward Euler discretization to the continuous-time dynamics, a proximal point-type iterative algorithm is derived, and its fixed-time convergence property is rigorously analyzed. Numerical experiments are provided to illustrate the effectiveness and advantages of the proposed methods.