Hessian-Riemannian second-order neurodynamic approach for constrained optimization: Theory and applications

Recently, Riemannian geometry has gained significant prominence as a foundational tool for the design and analysis of continuous trajectory flows, providing deep insights that fuel the development of novel methods for tackling nonlinear optimization problems. In this study, we introduce a second-order time-varying neurodynamic system with Hessian-Riemannian metric to solve a class of (possibly nonconvex) optimization problems, where the Hessian-Riemannian metric is derived from the Legendre-type function. Further, the proposed techniques and analyses also result in a novel approach for addressing optimization problems on Riemannian manifolds. First, we establish the existence and uniqueness of the system's solution. Subsequently, we examine the long-term behavior of the proposed system, at the same time, stability analysis for the system is done utilizing a strict Lyapunov function. Then, we analyze the convergence of the system in the continuous case, where the convergence results and rates are derived based on the Kurdyka-Łojasiewicz gradient inequality. These results are further extended to the discrete case, yielding similar conclusions. Finally, we validate the proposed method through some numerical examples, demonstrating its superior performance in simulations involving nonconvex test functions minimization, sparse signal recovery, and image deblurring.