Some remarks on conjugate gradient methods without line search

The conjugate gradient method is widely used in unconstrained optimization, especially in case of large-scale problems. However, the line search in the conjugate gradient method is sometimes very difficult or prohibitively expensive. In Sun and Zhang [J. Sun, J.P. Zhang, Global convergence of conjugate gradient methods without line search, Annals of Operations Research 103 (2001) 161–173], it is shown that by taking a “fixed” steplength αk defined by the formula αk=-δgkTdkdkTQkdk, the conjugate gradient method is globally convergent for several popular choices of βk without line search. In the simplest case all Qk could be identity matrices. However, it would not even guarantee the descent property. In this paper, we study some methods to select Qk, which are based on the amount of descent and are superior to taking Qk ≡ I (the unit matrix).