Wolfgang Quapp

A valley following method 

 Abstract

We present a procedure to follow the "path along the valley floor"
of a hypersurface. The aim is either to find minima, or to go from a
minimum to a saddle point of index one, if the saddle is at the top of
the valley floor.
The motivation is that of taking into account local nonconvexity of the
hypersurface and possibly to determine valleys.  
The method uses a projector technique where the projector is built by 
the tangent of the valley floor line. The projector is applied to the 
gradient and Hessian matrix of a given function, and it is used for
predictor and corrector steps in pathfollowing.
The resulting path is the "valley floor gradient extremal" which 
corresponds to the smallest (absolute) eigenvalue of the Hessian. 
Convergence properties are analysed.