A Linear Hybridization of the Hestenes–Stiefel Method and the Memoryless BFGS Technique
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We suggest a linear combination of search directions of the Hestenes–Stiefel method and the memoryless BFGS (Broyden–Fletcher–Goldfarb–Shanno) technique. As a result, a one-parameter extension of the Hestenes–Stiefel method is proposed. Based on an eigenvalue analysis, we show that the method may ensure the descent property. In a least-squares scheme, parameter of the method is determined in a way to tend the search direction of the method to the search direction of the three-term conjugate gradient method proposed by Zhang et al. which satisfies the sufficient descent condition. We conduct a brief global convergence analysis for the proposed method under the Wolfe line search conditions. Comparative numerical experiments are done on a set of the CUTEr test problems and the detailed results are reported. They show practical efficiency of the proposed method.
KeywordsNonlinear programming unconstrained optimization conjugate gradient method memoryless BFGS method global convergence
Mathematics Subject Classification90C53 65K05 65F35
This research was supported by Research Councils of Semnan University and Ferdowsi University of Mashhad. The authors thank the anonymous reviewer for his/her valuable comments helped to improve the presentation.
- 2.Andrei, N.: Numerical comparison of conjugate gradient algorithms for unconstrained optimization. Stud. Inform. Control 16(4), 333–352 (2007)Google Scholar
- 16.Dolan, E.D., Moré, J.J.: Benchmarking optimization software with performance profiles. Math. Program. 91(2, Ser. A):201–213 (2002)Google Scholar
- 18.Gould, N., Scott, J.: A note on performance profiles for benchmarking software. ACM Trans. Math. Softw. 43(2):Art. 15, 5 (2016)Google Scholar
- 25.Powell, M.J.D.: Nonconvex minimization calculations and the conjugate gradient method. In Griffiths, D.F., (Ed.), Numerical Analysis (Dundee, 1983), volume 1066 of Lecture Notes in Math., pp. 122–141. Springer, Berlin (1984)Google Scholar