The continuous Newton-Raphson method can look ahead

Abstract

This paper is about an intriguing property of the continuous Newton-Raphson method for the minimization of a continuous objective function f: if x is a point in the domain of attraction of a strict local minimizer x* then the flux line of the Newton-Raphson flow that starts in x approaches x* from a direction that depends only on the behavior of f in arbitrarily small neighborhoods around x and x*. In fact, if F is a sufficiently benign perturbation of f on an open region D not containing x, then the two flux lines through x defined by the Newton-Raphson vector fields that correspond to f and F differ from one another only within D.\ud \ud The work was supported by EPSRC grant GR/S34472 (R. Hauser) and by the Clarendon Fund, Oxford University Press and ORS Award, Universities UK (J Nedic