Abstract
A new modified Barzilai–Borwein gradient method for solving the strictly convex quadratic minimization problem is proposed by properly changing the Barzilai–Borwein stepsize such that some certain multi-step quasi-Newton condition is satisfied. The global convergence is analyzed. Numerical experiments show that the new method can outperform some known gradient methods.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Barzilai, J., Borwein, J.: Two-point step size gradient methods. IMA J. Numer. Anal. 8(1), 141–148 (1988)
Raydan, M., Svaiter, B.: Relaxed steepest descent and Cauchy–Barzilai–Borwein method. Comput. Optim. Appl. 21(2), 155–167 (2002)
Dai, Y., Liao, L.: R-linear convergence of the Barzilai and Borwein gradient method. IMA J. Numer. Anal. 22(1), 1–10 (2002)
Dai, Y., Yuan, Y.: Alternate minimization gradient method. IMA J. Numer. Anal. 23, 377–393 (2003)
Raydan, M.: On the Barzilai and Borwein choice of steplength for the gradient method. IMA J. Numer. Anal. 13(3), 321–326 (1993)
Dai, Y.: Alternate step gradient method. Optimization 52(4–5), 395–415 (2003)
Yuan, Y.: A new stepsize for the steepest descent method. J. Comput. Math. 24(2), 149–156 (2006)
De Asmundis, R., Di Serafino, D., Riccio, F., Toraldo, G.: On spectral properties of steepest descent methods. IMA J. Numer. Anal. 33(4), 1416–1435 (2013)
Gonzaga, C., Schneider, R.: On the steepest descent algorithm for quadratic functions. Comput. Optim. Appl. 63(2), 523–542 (2016)
Ford, J., Moghrabi, I.: Alternative parameter choices for multi-step quasi-Newton methods. Optim. Methods Softw. 2(3–4), 357–370 (1993)
Ford, J., Moghrabi, I.: Multi-step quasi-Newton methods for optimization. J. Comput. Appl. Math. 50(93), 305–323 (1994)
Dai, Y., Fletcher, R.: New algorithms for singly linearly constrained quadratic programs subject to lower and upper bounds. Math. Progr. 106(3), 403–421 (2006)
Friedlander, A., Martínez, J., Molina, B., Raydan, M.: Gradient method with retards and generalizations. SIAM J. Numer. Anal. 36(1), 275–289 (1999)
Acknowledgments
The authors are grateful to the anonymous referees and editor for their useful comments. This research was supported by the National Science Foundation of China under Grant Nos. 11171371 and 11571004.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Zheng, Y., Zheng, B. A New Modified Barzilai–Borwein Gradient Method for the Quadratic Minimization Problem. J Optim Theory Appl 172, 179–186 (2017). https://doi.org/10.1007/s10957-016-1008-9
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10957-016-1008-9