Exploiting Negative Curvature Directions 
       in Linesearch Methods for Unconstrained Optimization

     N. I. M. Gould   S. Lucidi   M. Roma   Ph. L. Toint

                        Report 97-18

In this paper we consider the definition of new efficient linesearch
algorithms   for  solving large  scale   unconstrained  optimization
problems  which  exploit the local   nonconvexity  of the  objective
function.  Existing algorithms of     this class compute,  at   each
iteration, two  search  directions: a   Newton-type direction  which
ensures a global  and  fast convergence,  and  a negative  curvature
direction which enables  the iterates to  escape from the region  of
local  nonconvexity. A new point is   then generated by performing a
movement along a  curve obtained by  combining these two directions.
However, the  respective  scaling  of the  directions  is  typically
ignored.  We propose a new algorithm which aims to avoid the scaling
problem by selecting  the more promising  of the two directions, and
then performs a step along this direction. The selection is based on
a test  on the  rate  of decrease   of the  quadratic model  of  the
objective function.  We   prove global convergence to   second-order
critical points  for the new algorithm,  and report some preliminary
numerical results.