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 $κ=L/μ$, where $μ$ is the strong-convexity constant and $L$ is the gradient Lipschitz constant. HNAG$^+$ attains the optimal global rate $1-2/\sqrtκ$, matching the information-theoretic lower bound. For functions with local asymptotic symmetry at the minimizer, HNAG$^{++}$ attains the asymptotic rate $1-2\sqrt{2/κ}$. This matches the best known asymptotic rate under $\mathcal C^2$ regularity, while applying to a broader function class. Numerical experiments confirm the predicted rates and show favorable performance against existing accelerated schemes.