Abstract
Using Taylor’s series, we propose a modified secant relation to get a more accurate approximation of the second curvature of the objective function. Then, using this relation and an approach introduced by Dai and Liao, we present a conjugate gradient algorithm to solve unconstrained optimization problems. The proposed method makes use of both gradient and function values, and utilizes information from the two most recent steps, while the usual secant relation uses only the latest step information. Under appropriate conditions, we show that the proposed method is globally convergent without needing convexity assumption on the objective function. Comparative results show computational efficiency of the proposed method in the sense of the Dolan–Moré performance profiles.



Similar content being viewed by others
References
Al-Baali M (1985) Descent property and global convergence of the Fletcher- Reeves method with inexact linesearch. IMA J. Numer. Anal. 5:121–124
Babaie-Kafaki S, Ghanbari R (2014) The Dai-Liao nonlinear conjugate gradient method with optimal parameter choices. Eur. J. Oper. Res. 234:625–630
Babaie-Kafaki S, Ghanbari R, Mahdavi-Amiri N (2010) Two new conjugate gradient methods based on modified secant relations. J. Comput. Appl. Math. 234:1374–1386
Dai YH, Han JY, Liu GH, Sun DF, Yin HX, Yuan YX (1999) Convergence properties of nonlinear conjugate gradient methods. SIAM J. Optim. 10:348–358
Dai YH, Liao LZ (2001) New conjugacy conditions and related nonlinear conjugate gradient methods. Appl. Math. Optim. 43:87–101
Dolan ED, Moré JJ (2002) Benchmarking optimization software with performance profiles. Math. Program. 91:201–213
Fletcher R, Revees CM (1964) Function minimization by conjugate gradients. Comput. J. 7:149–154
Ford JA, Moghrabi IA (1997) Alternating multi-step quasi-Newton methods for unconstrained optimization. J. Comput. Appl. Math. 82:105–116
Ford JA, Moghrabi IA (1993) Alternative parameter choices for multi-step quasi-Newton methods. Optim. Methods Softw. 2:357–370
Ford JA, Moghrabi IA (1994) Multi-step quasi-Newton methods for optimization. J. Comput. Appl. Math. 50:305–323
Ford JA, Narushima Y, Yabe H (2008) Multi-step nonlinear conjugate gradient methods for unconstrained minimization. Comput. Opt. Appl. 40:191–216 21-42 (1992)
Gould NI, Orban D, Toint PhL (2015) CUTEst: a constrained and unconstrained testing environment with safe threads for mathematical optimization. Comput. Opt. Appl. 60:545–557
Hestenes MR, Stiefel EL (1952) Methods of conjugate gradients for solving linear systems. J. Res. Nat. Bur. Stand. 49:409–436
Li G, Tang C, Wei Z (2007) New conjugacy condition and related new conjugate gradient methods for unconstrained optimization. J. Comput. Appl. Math. 202:523–539
Moré JJ, Thuente DJ (1994) Line search algorithms with guaranteed sufficient decrease. ACM Trans. Math. Softw. 20:286–307
Moghrabi IA (2019) A new preconditioned conjugate gradient method for optimization. IAENG Int. J. Appl. Math. 49(1):1–8
Nocedal J, Wright SJ (2006) Numerical Optimization. Springer, New York
Polak E, Ribiére G (1969) Note Sur la Convergence de Directions Conjuguée. Francaise Informat Recherche Operationelle 16:35–43
Polyak BT (1969) The conjugate gradient method in extreme problems. USSR Comput. Math. Math. Phys. 9:94–112
Powell MJD (1977) Restart procedures of the conjugate gradient method. Math. Program. 2:241–254
Powell MJD (1984) Nonconvex minimization calculations and the conjugate gradient method, Numerical Analysis (Dundee, 1983) Lecture Notes in Mathematics, vol 1066. Springer, Berlin, pp 122–141
Wei Z, Li G, Qi L (2006) New quasi-Newton methods for unconstrained optimization problems. Appl. Math. Comput. 175:1156–1188
Yuan G, Wei Z (2010) Convergence analysis of a modified BFGS method on convex minimizations. Comput. Opt. Appl. 47:237–255
Yabe H, Takano M (2004) Global convergence properties of nonlinear conjugate gradient methods with modified secant relation. Comput. Optim. Appl. 28:203–225
Zhang JZ, Xu CX (2001) Properties and numerical performance of quasi-Newton methods with modified quasi-Newton equation. J. Comput. Appl. Math. 137:269–278
Zoutendijk G (1970) Nonlinear programming, computational methods. In: Abadie J (ed) Integer and nonlinear programming. North-holland, Amsterdam, pp 37–86
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Andreas Fischer.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Dehghani, R., Bidabadi, N. Two-step conjugate gradient method for unconstrained optimization. Comp. Appl. Math. 39, 241 (2020). https://doi.org/10.1007/s40314-020-01297-2
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s40314-020-01297-2