A descent family of the spectral Hestenes–Stiefel method by considering the quasi-Newton method

The prominent computational features of the Hestenes–Stiefel parameter as one of the fundamental members of conjugate gradient methods have attracted the attention of many researchers. Yet, as a weak stop, it lacks global convergence for general functions. To overcome this defect, a family of spectral version of Hestenes–Stiefel conjugate gradient methods is introduced. To compute the spectral parameter, in the account of worthy properties of quasi-Newton methods, we minimize the distance between the search direction matrix of the spectral conjugate gradient method and the BFGS (Broyden–Fletcher–Goldfarb–Shanno) update. To achieve sufficient descent property, the search direction is projected in the orthogonal subspace to the gradient of the objective function. The convergence analysis of the proposed method is carried out under standard assumptions for general functions. Finally, the practical merits of the suggested method are investigated by numerical experiments on a set of CUTEr test functions using the Dolan–Moré performance profile. The results show the computational efficiency of the proposed method.