This paper is focused on improving global convergence of the modified BFGS algorithm with Yuan-Wei-Lu line search formula. This improvement has been achieved by presenting a different line search approach and it is proved that the BFGS method with this line search converges globally if the function to be minimized has Lipschitz continuous gradients. The performance of the suggested algorithm is investigated via mathematical analysis and a simulation study.
This is a preview of subscription content, log in to check access.
Buy single article
Instant access to the full article PDF.
Price includes VAT for USA
Subscribe to journal
Immediate online access to all issues from 2019. Subscription will auto renew annually.
This is the net price. Taxes to be calculated in checkout.
Fu, Z., Ren, K., Shu, J., Sun, X., Huang, F.: Enabling personalized search over encrypted outsourced data with efficiency improvement. IEEE Trans. Parallel Distrib. Syst. 27(9), 2546–2559 (2016)
Xia, Z., Wang, X., Zhang, L., Qin, Z., Sun, X., Ren, K.: A privacy-preserving and copy-deterrence content-based image retrieval scheme in cloud computing. IEEE Trans. Inf. Forensic. Secur. 11(11), 2594–2608 (2016)
Broyden, C.G.: The convergence of a class of double-rank minimization algorithms 1. general considerations. IMA J. Appl. Math. 6(1), 76–90 (1970)
Fletcher, R.: A new approach to variable metric algorithms. Comput. J. 13 (3), 317–322 (1970)
Goldfarb, D.: A family of variable-metric methods derived by variational means. Math. Comput. 24(109), 23–26 (1970)
Shanno, D.F.: Conditioning of quasi-newton methods for function minimization. Math. Comput. 24(111), 647–656 (1970)
Powell, M.J.: Some global convergence properties of a variable metric algorithm for minimization without exact line searches. Nonlinear Programm. 9(1), 53–72 (1976)
Andrei, N.: A double parameter scaled BFGS method for unconstrained optimization. J. Comput. Appl. Math. 332, 26–44 (2018)
Broyden, C.G., Dennis, J. Jr, Moré, J.J.: On the local and superlinear convergence of quasi-Newton methods. IMA J. Appl. Math. 12(3), 223–245 (1973)
Dennis, J.E., Moré, J.J.: A characterization of superlinear convergence and its application to quasi-Newton methods. Math. Comput. 28(126), 549–560 (1974)
Griewank, A., Toint, P.L.: Local convergence analysis for partitioned quasi-Newton updates. Numer. Math. 39(3), 429–448 (1982)
Dai, Y.-H.: Convergence properties of the BFGS algorithm. SIAM J. Optim. 13(3), 693–701 (2002)
Yuan, G., Wei, Z., Lu, X.: Global convergence of BFGS and PRP methods under a modified weak wolfe–powell line search. Appl. Math. Model. 47, 811–825 (2017)
Yuan, G., Sheng, Z., Wang, B., Hu, W., Li, C.: The global convergence of a modified BFGS method for nonconvex functions. J. Comput. Appl. Math. 327, 274–294 (2018)
Nocedal, J., Wright, S.J.: Numerical optimization 2nd (2006)
Byrd, R.H., Nocedal, J.: A tool for the analysis of quasi-Newton methods with application to unconstrained minimization. SIAM J. Numer. Anal. 26(3), 727–739 (1989)
Li, D.H., Fukushima, M.: On the global convergence of the BFGS method for nonconvex unconstrained optimization problems. SIAM J. Optim. 11(4), 1054–1064 (2001)
Andrei, N.: An unconstrained optimization test functions collection. Adv. Model. Optim. 10(1), 147–161 (2008)
Yuan, Y., Sun, W., et al.: Theory and Methods of Optimization. Science Press of China, Beijing (1999)
Dolan, E.D., Moré, J. J.: Benchmarking optimization software with performance profiles. Math. Programm. 91(2), 201–213 (2002)
Polyak, B.T.: The conjugate gradient method in extremal problems. USSR Comput. Math. Math. Phys. 9(4), 94–112 (1969)
Polak, E., Ribiere, G.: Note sur la convergence de méthodes de directions conjuguées, Revue franċaise D’informatique et de Recherche opérationnelle. Sér. Rouge 3(16), 35–43 (1969)
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
About this article
Cite this article
Hosseini Dehmiry, A. The global convergence of the BFGS method under a modified Yuan-Wei-Lu line search technique. Numer Algor 84, 781–793 (2020). https://doi.org/10.1007/s11075-019-00779-7
- Quasi-Newton method
- BFGS method
- Global convergence
- Unconstrained optimization
Mathematics Subject Classification (2010)