Some new three-term Hestenes–Stiefel conjugate gradient methods with affine combination

Abstract

In this paper, a modified Hestenes–Stiefel conjugate gradient method for unconstrained problems is developed, which can achieves the twin goals of generating sufficient descent direction at each iteration as well as being close to the Newton direction. In our methods, the hybridization parameter can also be obtained based on other kinds of conjugacy conditions. Under mild condition, we establish their global convergence for general objective functions. Numerical experimentation with the new method indicates that it efficiently solves the test problems and therefore is promising.

Keywords:

• Three-term conjugate gradient method
• sufficient descent condition
• quasi-Newton condition
• global convergence
• affine combination

Acknowledgements

We are sincerely grateful to the anonymous referees and editor for their many constructive and valuable suggestions and comments, which have made the paper clearer and more comprehensive than the earlier version. We would like to thank Professors W.W.Hager and H. Zhang for their CG_DESCENT code for numerical comparison.

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work is supported by the Natural Science Foundation of China [grant number 11601012]; [grant number 71271033]; the Scientific Research Foundation of the Higher Education Institutions of Ningxia [grant number NGY2016134], Beifang University of Nationalities Foundation [grant number 2016SXKY06], Scientific Research Fund of Hunan Provincial Education Department [grant number 16B005], Guangxi Key Laboratory of Automatic Detecting Technology and Instruments [grant number YQ16112], Guangxi Natural Science Foundation [grant number 2014GXNSFAA118003].