The Continuous Newton--Raphson Method Can Look Ahead

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 $\hat{f}$ 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 $\hat{f}$ differ from one another only within D.