Two efficient spectral hybrid CG methods based on memoryless BFGS direction and Dai–Liao conjugacy condition

Abstract

The spectral conjugate gradient (CG) method is one of the effective methods for solving unconstrained optimization problems. In this work, we introduce a composite hybrid CG parameter which is a convex combination of two new adaptive hybrid CG parameters. To derive the optimal choice of the combination coefficient in our proposed composite CG parameter, two approaches to calculating it are introduced. One is to minimize the distance between the hybrid CG direction and the self-scaling memoryless BFGS direction and the other is to apply the Dai–Liao conjugacy condition. Further, to make the search direction have better theoretical performance, two effective spectral hybrid CG methods are generated. Our proposed methods ensure the sufficient descent property regardless of the line search. And the global convergence results for general non-convex functions are established under some fundamental assumptions and Wolfe line search. Numerical experiments on solving unconstrained optimization problems illustrate the effectiveness of our proposed methods. 

Keywords:

• Unconstrained optimization
• spectral hybrid conjugate gradient method
• memoryless BFGS direction
• Dai–Liao conjugacy condition
• global convergence

2020 Mathematics Subject Classifications:

• 90C25
• 90C30

Acknowledgments

The authors are grateful to the anonymous referees for their constructive comments and suggestions to improve the quality and clarity of the paper.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 Their codes can be downloaded at https://github.com/jhyin-optim/FHTTCGMs_with_applications.

Additional information

Funding

This work is supported by the National Natural Science Foundation of China [grant number 72071202] and Key Laboratory of Mathematics and Engineering Applications, Ministry of Education.

Notes on contributors

Pengjie Liu

Pengjie Liu received an M.S. degree from Guangxi University in 2021. He is pursuing a PhD in the School of Mathematics at China University of Mining and Technology. His research focuses on transportation network modelling and the analysis of optimization algorithm.

Zihang Yuan

Zihang Yuan is currently pursuing a bachelor's degree in mathematics at the School of Mathematics, China University of Mining and Technology. His research interests include the analysis of optimization algorithm.

Yue Zhuo

Yue Zhuo received her M.S. degree from China University of Mining and Technology in 2023 and she is currently pursuing a Ph.D. at the same institution. Her research interests include transportation network modelling and the analysis of optimization algorithm.

Hu Shao

Hu Shao received his PhD degree from Nanjing University. He is a professor in the School of Mathematics at China University of Mining and Technology. He mainly focuses on transportation network modelling, traffic big data analysis and modelling, intelligent transportation systems and analysis of optimization algorithm.