A note on Branin’s method for finding the critical points of smooth functions. (English) Zbl 0625.65067

Parametric optimization and related topics, Int. Conf., Plaue/GDR 1985, Math. Res. 35, 209-228 (1987).
[For the entire collection see Zbl 0619.00012.]
For a generic subclass of smooth functions f on \({\mathbb{R}}^ n\), we study a globally defined autonomous differential equation on \({\mathbb{R}}^ n\), underlying the Newton-iteration method in the continuous version of F. H. Branin jun. [IBM J. Res. Develop. 16, 504-522 (1972; Zbl 0271.65034)] for finding critical points of f. We describe the set of all equilibria of the differential equation and discuss the local phase portraits around these points.


65L05 Numerical methods for initial value problems involving ordinary differential equations
65H10 Numerical computation of solutions to systems of equations
65K05 Numerical mathematical programming methods
37G99 Local and nonlocal bifurcation theory for dynamical systems