A dynamical system for solving inverse quasi-variational inequalities

Abstract

We study the existence and uniqueness of solutions to the inverse quasi-variational inequality problem. Motivated by the dynamical approach to solving optimization problems such as variational inequality, monotone inclusion, and inverse variational problems, we consider a dynamical system associated with the inverse quasi-variational inequality problem, and establish the existence and uniqueness of a solution to the proposed system. We prove that every trajectory of the proposed dynamical system converges to the unique solution of the inverse quasi-variational inequality problem and that the system is globally asymptotically stable at its equilibrium point. We also prove that if the function which governs the inverse quasi-variational inequality problem is strongly monotone and Lipschitz continuous, then the dynamical system is globally exponentially stable at its equilibrium point. We discretize the dynamical system and show that the sequence generated by the discretization of the system converges strongly to the unique solution of the inverse quasi-variational inequality problem under certain assumptions on the parameters involved. Finally, we provide numerical examples to support and illustrate our theoretical results.

Keywords:

• Convergence
• dynamical system and global stability
• inverse quasi-variational inequality
• monotone mapping
• projection

AMS CLASSIFICATIONS:

• 65P40
• 47J20
• 49J40

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The first author gratefully acknowledges the financial support of the Post-Doctoral Program at the Technion - Israel Institute of Technology. The second author was partially supported by the Israel Science Foundation (Grant No. 820/17), by the Fund for the Promotion of Research at the Technion and by the Technion General Research Fund.