Scaled projection neural network for quasi-variational inequalities.

We propose a scaled projection neural network for solving quasi-variational inequalities in which the constraint set S(x)=m(x)+S depends on the state through a contraction mapping m that translates a fixed closed convex set S, while the projection is performed with respect to a fixed symmetric positive definite matrix M. The matrix M acts as a continuous-time preconditioner, reshaping the geometry of the projected flow without altering the solution set. We establish that equilibria of the proposed dynamics coincide exactly with solutions of the quasi-variational inequality, and prove well-posedness under Lipschitz regularity of the operator and the translation mapping. A stability analysis based on a weighted Lyapunov function induced by the inverse of M yields an explicit contraction parameter that captures the interplay between the matrix bounds and the contraction properties of the translation, generalizing classical step-size conditions for both fixed-set variational inequalities and quasi-variational inequalities. Under an explicit dominance condition relating the monotonicity modulus, the Lipschitz constants, and the matrix bounds, we prove global exponential convergence on a computable step-size window, and a forward-Euler discretization preserves the certified rate. Numerical experiments across five application domains illustrate the role of M as a preconditioner and exhibit convergence in regimes where the certificate is silent. The framework connects and extends two lines of research that have not been previously combined: Euclidean projected dynamical systems for quasi-variational inequalities and state-dependent matrix projection neural networks for variational inequalities with fixed constraint sets.