$$\hbox {HNAG}^{++}$$: An Accelerated Gradient Method with a Refined Asymptotic Rate for Strongly Convex Optimization

Abstract Two accelerated first-order methods, $$\hbox {HNAG}^+$$ HNAG + and $$\hbox {HNAG}^{++}$$ 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 $$\kappa =L/\mu $$ κ = L / μ , where $$\mu $$ μ is the strong-convexity constant and L is the gradient Lipschitz constant. To leading order as $$\kappa \rightarrow \infty $$ κ → ∞ , $$\hbox {HNAG}^+$$ HNAG + attains the global rate $$1-2/\sqrt{\kappa }$$ 1 - 2 / κ , matching the information-theoretic lower bound. Under local asymptotic symmetry at the minimizer, $$\hbox {HNAG}^{++}$$ HNAG + + achieves the faster asymptotic rate $$1-2\sqrt{2/\kappa }$$ 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 $${\mathcal {C}}^2$$ 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.