Termination and equivalence results for conjugate gradient algorithms

Abstract

This paper considers the problem of minimizing a functionf(x) when linear equality constraints onx could be present and when an algorithm is to be used which may employ both conjugate gradient and quasi-Newton steps. The conjugate gradient algorithm is examined when a preconditioner is used which may be only positive semi-definite. Quite a general theorem is given about the equivalence between conjugate gradient steps and quasi-Newton steps which includes several previously published results as special cases. Some results are also given for the case where some line searches are inexact.