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 f̂ 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 f̂ differ from one another only within D.