Tseng's Type Methods in Continuous and Discrete Time for Quasi-Variational Inequalities
Lkhamsuren Altangerel
TL;DR
This work addresses quasi-variational inequalities (QVIs) in real Hilbert spaces, where the constraint set depends on the solution, by extending Tseng's scheme to both continuous and discrete time. It develops a Tseng-type dynamical system and a Tseng-type projection scheme with moving sets, proving existence and uniqueness of solutions under $F$ being $L$-Lipschitz and $\rho$-strongly monotone, and establishing exponential or linear convergence rates under explicit parameter conditions involving the projection-continuity constant $l$ and $\theta = l + \sqrt{1 - 2\lambda\rho + \lambda^2 L^2}$. The paper also provides a concrete $\ell_2$-space example with a moving-set $K(x)$ and discusses a special case $K(x)=m(x)+K$ that leads to a relaxed bound $l \le 2\beta$, yielding adjusted convergence criteria. Overall, the results offer practical, provable algorithms for solving QVIs with solution-dependent constraints and highlight directions for extending the framework to weaker monotonicity assumptions and broader moving-set structures.
Abstract
This paper presents an approach for obtaining approximate solutions to quasi-variational inequalities in a real Hilbert space by modifying Tseng's scheme, which was originally designed for variational inequalities. The study explores the existence of equilibrium points and investigates convergence results related to dynamical systems. Linear convergence for discretized systems is examined through examples, illustrations, and special cases.