HNAG++: An Accelerated Gradient Method with a Refined Asymptotic Rate for Strongly Convex Optimization

Two accelerated first-order methods, HNAG^+ and HNAG++, are introduced for smooth strongly convex optimization. They are derived from the Hessian-driven Nesterov Accelerated Gradient (HNAG) flow by optimizing the coercivity of shifted Lyapunov functions. Let kappa=L/μ, where μ is the strong-convexity constant and L is the gradient Lipschitz constant. HNAG^+ attains the optimal global rate 1-2/√kappa, matching the information-theoretic lower bound. For functions with local asymptotic symmetry at the minimizer, HNAG++ attains the asymptotic rate 1-2√2/kappa. This matches the best known asymptotic rate under mathcal C2 regularity, while applying to a broader function class. Numerical experiments confirm the predicted rates and show favorable performance against existing accelerated schemes.