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

Two accelerated first-order methods, HNAG⁺ HNAG + and HNAG⁺⁺ 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 κ = L / μ , where μ μ is the strong-convexity constant and L is the gradient Lipschitz constant. To leading order as κ → ∞ κ → ∞ , HNAG⁺ HNAG + attains the global rate 1-2/√κ 1 - 2 / κ , matching the information-theoretic lower bound. Under local asymptotic symmetry at the minimizer, HNAG⁺⁺ HNAG + + achieves the faster asymptotic rate 1-2√2/κ 1 - 2 2 / κ to leading order. This matches the fastest known asymptotic rate for twice continuously differentiable objectives at leading order and extends it to a strictly larger function class. In particular, local asymptotic symmetry permits objectives that are not twice differentiable at the minimizer, showing that C² C 2 regularity is not necessary for this accelerated rate. A unified Lyapunov analysis establishes these guarantees, and numerical experiments confirm the predicted rates and show favorable performance against existing accelerated schemes.