W.Quapp

Computers and Mathem. with Applic. 41 No. 3/4, (2001) 407-414 

see: http://www.elsevier.nl/locate/camwa

"Searching Minima of an N-dimensional Surface: 
 A Robust Valley Following Method" 


Abstract
A procedure is proposed to follow the "minimum path" of a hypersurface
starting anywhere in the catchment region of the corresponding minimum. The
method uses a modification of the so-called "following the reduced
gradient". 
The original method connects points where the gradient has a constant
direction. In the present letter, this is replaced by the successive
directions of the tangent of the searched curve. The resulting pathway is
that valley floor gradient extremal which belongs to the smallest (absolute)
eigenvalue of the Hessian. The new method avoids third derivatives of the
objective function. The effectiveness of the algorithm is demonstrated by
using a polynomial test, the notorious Rosenbrock function in two, 20, and
in 100 dimensions.

Keywords: Minima; Path following; Saddle point; Reduced gradient; 
          Gradient extremal.