  This paper introduces the Runge-Kutta Chebyshev descent method (RKCD) for
strongly convex optimisation problems. This new algorithm is based on explicit
stabilised integrators for stiff differential equations, a powerful class of
numerical schemes that avoid the severe step size restriction faced by standard
explicit integrators. For optimising quadratic and strongly convex functions,
this paper proves that RKCD nearly achieves the optimal convergence rate of the
conjugate gradient algorithm, and the suboptimality of RKCD diminishes as the
condition number of the quadratic function worsens. It is established that this
optimal rate is obtained also for a partitioned variant of RKCD applied to
perturbations of quadratic functions. In addition, numerical experiments on
general strongly convex problems show that RKCD outperforms Nesterov's
accelerated gradient descent.
