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.
This paper proposes a bilevel particle swarm optimization (PSO)-neurodynamic method for solving the distributed sparse least-squares problem regularized by the smoothly clipped absolute deviation (SCAD) penalty to global optimality. The SCAD penalty is nonsmooth and nonconvex, so existing distributed neurodynamic methods have not been extended to it directly. Through two-time-scale singular perturbation analysis, a rigorous full-trajectory convergence theory with explicit convergence rates is established for the inner collaborative multi-agent neurodynamic system without requiring the second-order sufficient condition. A theory-guided inner-outer collaborative optimization framework is further developed, in which a red-light green-light sensitivity mechanism is embedded in the outer PSO search to enhance global search while concentrating computational effort on the SCAD shape parameters, thereby improving recovery accuracy and computational efficiency. Simulation results on four sparse-recovery scenarios demonstrate that the proposed method accurately recovers the true sparsity pattern and achieves faster convergence and lower relative error than competing neurodynamic methods.