A note on the second-order convergence of optimization
            algorithms using barrier functions

         Nicholas I. M. Gould and Philippe L. Toint

                      Report 97-15 


It has long been known that barrier algorithms for constrained
optimization can produce  a sequence of iterates converging to
a   critical point  satisfying   weak  second-order  necessary
optimality conditions,  when  their inner  iterations  ensures
that  second-order  necessary conditions hold  at each barrier
minimizer.  We  show  that, despite  this, strong second-order
necessary   conditions may fail to  be  attained at the limit,
even if the barrier minimizers satisfy second-order sufficient
optimality conditions.