Memoryless Quasi-Newton Methods

In Sect. 2.10 we have presented the conjugate gradient algorithm derived from the Hestenes-Stiefel method. The method is globally convergent under the Wolfe line search rules. Its strong convergent properties are achieved by modifying the coefficient β _k in such a way that it is no longer equivalent to the nonlinear Hestenes-Stiefel conjugate gradient algorithm. In Sect. 3.2 we show that the nonlinear Polak-Ribière method is equivalent to the nonlinear Hestenes-Stiefel algorithm provided that the directional minimization is exact. Having that in mind and the fact that Hager and Zhang do not stipulate condition (2.68) in Theorem 2.14 their main convergence result is remarkable.