Global convergence result for conjugate gradient methods

Conjugate gradient optimization algorithms depend on the search directions, $\begin{gathered} s^{(1)} = - g^{(1)} , \hfill s^{(k + 1)} = - g^{(k + 1)} + \beta ^{(k)} s^{(k)} ,k \geqslant 1, \hfill \end{gathered} with different methods arising from different choices for the scalar β(k). In this note, conditions are given on β(k) to ensure global convergence of the resulting...