Acceleration in convex optimization via Runge-Kutta discretization of high-resolution ordinary differential equations

This paper investigates first-order optimization algorithms derived by discretizing two high-resolution ordinary differential equations. For sufficiently smooth objective functions, we demonstrate that acceleration can be achieved via Runge-Kutta integrators. Specifically, for convex objective functions f satisfying the assumptions of Lipschitz continuous gradients and (s+2)-th order differentiability, we prove that the objective function converges to its minimum value with a rate of O(N−2ss+1), where s denotes the order of the Runge-Kutta integrator. For strongly convex objective functions f, under the same assumptions, we further prove that the proposed algorithms achieve a faster convergence rate than the algorithms generated by symplectic Euler discretization. Numerical experiments validate the effectiveness and superiority of the proposed algorithms through comparisons with classical benchmark methods, including Nesterov's accelerated gradient method and the symplectic Euler scheme.