, Volume 71, Issue 2, pp 399-405

Global convergence result for conjugate gradient methods

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Abstract

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 algorithms.

Communicated by L. C. W. Dixon