Convergence Rates for Variational Inequality Projection Neural Networks with a State-Dependent Metric

Abstract

We study continuous-time projection neural networks for variational inequalities on closed convex sets. A positive definite matrix that depends on the state preconditions the operator, and its inverse defines the projection metric. Existing convergence analyses of this flow cover Hessian-generated inverse metrics and state-dependent scalar metrics. In the first case, a Bregman distance eliminates metric-derivative terms. We treat a matrix metric whose inverse is of neither kind. We prove joint regularity of the projection in its argument and metric. Under common spectral bounds, the Euclidean Lipschitz estimate improves from the squared bound to the bound itself. For Lipschitz strongly monotone operators and Lipschitz metrics, we prove local exponential convergence with explicit rate and radius. On compact feasible sets, an explicit metric-variation bound yields global exponential convergence. A twice continuously differentiable, uniformly positive definite metric and a strongly monotone linear operator produce an annulus of periodic orbits. The inverse metric violates Hessian integrability throughout the annulus. Deterministic computations confirm the analytic formulas and quantify slack in both convergence certificates.