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

AbstractThe 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.Keywords: Nonlinear programmingspectral conjugate gradient methodquasi-Newton methodsufficient descent propertyglobal convergence AcknowledgementsThe authors thank the anonymous Reviewers and the Associate Editor for their valuable comments and suggestions that helped to improve the quality of this work. They are grateful to Professor Michael Navon for providing the line search code.Disclosure statementNo potential conflict of interest was reported by the author(s).Additional informationFundingThis research was supported by the Research Council of Semnan University.Notes on contributorsMaryam Khoshsimaye-BargardMaryam Khoshsimaye-Bargard is a Ph.D. candidate in the Department of Mathematics of Semnan University, Iran. She received her BSc in Applied Mathematics from Payame Noor University of Shahrood, Iran, in 2007 and her MSc in Operational Research from Shahrood University of Technology, Iran, in 2011. Her research interests are large-scale optimization, nonlinear programming and matrix computations.Ali AshrafiAli Ashrafi joined Semnan University in the year 2011. Currently, he serves as Associate Professor of Operational Research. He is a member of the Iranian Operational Research Society, and the Iranian Mathematical Society. His research interests include linear and non- linear optimization, and data envelopment analysis. He is the author of over 60 journal articles and conference proceedings.