Recently, important contributions on convergence studies of conjugate gradi
ent methods were made by Gilbert and Nocedal [SIAM J. Optim., 2 (1992), pp.
21-42]. They introduce a "sufficient descent condition" to establish globa
l convergence results. Although this condition is not needed in the converg
ence analyses of Newton and quasi-Newton methods, Gilbert and Nocedal hint
that the sufficient descent condition, which was enforced by their two-stag
e line search algorithm, may be crucial for ensuring the global convergence
of conjugate gradient methods. This paper shows that the sufficient descen
t condition is actually not needed in the convergence analyses of conjugate
gradient methods. Consequently, convergence results on the Fletcher-Reeves
- and Polak-Ribiere-type methods are established in the absence of the suff
icient descent condition.
To show the differences between the convergence properties of Fletcher-Reev
es- and Polak-Ribiere-type methods, two examples are constructed, showing t
hat neither the boundedness of the level set nor the restriction beta(k) gr
eater than or equal to 0 can be relaxed for the Polak-Ribiere-type methods.