A Second-Order Dynamical System for Solving Generalized Inverse Mixed Variational Inequality problems
Nam Van Tran
TL;DR
This work addresses generalized inverse mixed variational inequality problems (GIMVIPs) by introducing a projection-based second-order time-varying dynamical system and its discrete inertial counterpart. Under strong monotonicity and Lipschitz conditions on the underlying operators, the authors establish global existence, uniqueness, and global exponential convergence of the continuous trajectories to the unique GIMVIP solution, and derive a discretization with a linear convergence rate. The analysis hinges on the generalized $f$-projection and a residual operator $A(w)= T(w)-P_K^{\gamma f}(T(w)-\gamma g(w))$, with careful Lyapunov-based arguments to guarantee stability. Numerical experiments in a 1D setting confirm accelerated convergence of the inertial scheme relative to the first-order counterpart, demonstrating the practical viability of second-order dynamics for inverse variational problems.
Abstract
In this paper, we study a class of generalized inverse mixed variational inequality problems (GIMVIPs). We propose a novel projection-based second-order time-varying dynamical system for solving GIMVIPs. Under the assumptions that the underlying operators are strongly monotone and Lipschitz continuous, we establish the existence and uniqueness of solution trajectories and prove their global exponential convergence to the unique solution of the GIMVIP. Furthermore, a discrete-time realization of the continuous dynamical system is developed, resulting in an inertial projection algorithm. We show that the proposed algorithm achieves linear convergence under suitable choices of parameters. Finally, numerical experiments are presented to illustrate the effectiveness and convergence behavior of the proposed method in solving GIMVIPs.