A continuous Newton-type method for unconstrained optimization

In this paper, we propose a continuous Newton-type method in the form of an ordinary differential equation by combining the negative gradient and Newton’s direction. It is shown that for a general function f(x), our method converges globally to a connected subset of the stationary points of f(x) under some mild conditions; and converges globally to a single stationary point for a real analytic function. The method reduces to the exact continuous Newton method if the Hessian matrix of f(x) is positive definite. The convergence of the new method on the set of standard test problems in the literature are also reported.