Projection-Free Gradient Tracking Neurodynamic for Distributed Nonnegative Sparse Optimization

Abstract

This paper proposes a continuous-time, projection-free neurodynamic system based on gradient tracking for distributed nonnegative sparse optimization. Unlike traditional methods, our approach avoids projecting states onto the feasible set. Instead, it employs gradient tracking to estimate average gradients under heterogeneous costs, coupling primal and dual dynamics such that feasibility and sparsity emerge naturally from the equilibrium conditions. The system ensures unique solutions through a locally Lipschitz vector field and guarantees that every equilibrium corresponds to an exact KKT point. We establish rigorous convergence guarantees for convex objectives through three distinct analytical layers. These include a centralized kernel rate, a computable distributed rate derived from Jacobian analysis without restrictive stability assumptions, and a global exponential stability proof based on Lur'e system theory. Our analysis quantitatively separates these layers, revealing that the convergence rate is primarily limited by graph sparsity and remains robust to variations in momentum time scales. Numerical experiments on strongly convex quadratic problems validate the theoretical rates, demonstrate effective active set recovery, and show that the system maintains stable tail convergence rates even under strong local heterogeneity.