Abstract
To improve upon numerical stability of the spectral conjugate gradient methods, two adaptive scaling parameters are introduced. One parameter is obtained by minimizing an upper bound of the condition number of the matrix involved in producing the search direction and the other one is obtained by minimizing the Frobenius condition number of the matrix. The proposed methods are shown to be globally convergent, under appropriate conditions. A comparative testing of the proposed methods and some efficient spectral conjugate gradient methods shows the computational efficiency of the proposed methods.
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References
Birgin E, Martinez JM (2001) A spectral conjugate gradient method for unconstrained optimization. Appl Math Optim 43:117–128
Andrei N (2007) Scaled conjugate gradient algorithms for unconstrained optimization. Comput Optim Appl 38:401–416
Dehghani R, Mahdavi-Amiri N (2018) Scaled nonlinear conjugate gradient methods for nonlinear least squares problems. Numerical Algorithms 79:1–20
Hager WW, Zhang H (2006) Algorithm 851: CGDESCENT, a conjugate gradient method with guaranteed descent. ACM Trans Math Softw 32:113–137
Yu GH, Guan LT, Chen WF (2008) Spectral conjugate gradient methods with sufficient descent property for large-scale unconstrained optimization. Optim Methods Softw 23:275–293
Zhang Y, Dan B (2016) An efficient adaptive scaling parameter for the spectral conjugate gradient method. Optim Lett 10(1):119–136
Sun W, Yuan YX (2006) Optimization theory and methods: nonlinear programming. Springer, New York
Nocedal J, Wright SJ (2006) Numerical optimization. Springer, New York
Fletcher R (2005) On the Barzilai-Borwein method. In: Qi L, Teo K, Yang X (eds) Optimization and Control Applications, Series: Applied Optimization, vol 96. Springer Berlin, pp 35–256
Hager WW, Zhang H (2005) A new conjugate gradient method with guaranteed descent and an efficient line search. SIAM J Optim 16:170–192
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
Polak E, Ribiére G (1969) Note sur la convergence de directions conjugue. 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
Fletcher R, Revees CM (1964) Function minimization by conjugate gradients. Comput J 7:149–154
Raydan M (1997) The Barzilai and Borwein gradient method for the large scale unconstrained minimization problem. SIAM J Optim 7:26–33
Shanno DF, Phua KH (1976) Algorithm 500: minimization of unconstrained multivariate functions. ACM Trans Math Software 2:87–94
Gould NIM, Orban D, Toint PhL (2003) CUTEr, a constrained and unconstrained testing environment, revisited. ACM Trans Math Softw 29:373–394
Dolan ED, Moré JJ (2002) Benchmarking optimization software with performance profiles. Math Program 91:201–213
Hager WW, Zhang H (2006) A survey of nonlinear conjugate gradient methods. Pac J Optim 2:35–58
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The corresponding author thanks Strasbourg University and the other authors thank Lebanese International University and Yazd University for supporting this work.
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Fahs, A., Fahs, H. & Dehghani, R. Optimal Scaling Parameters for Spectral Conjugate Gradient Methods. Oper. Res. Forum 3, 31 (2022). https://doi.org/10.1007/s43069-022-00141-z
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DOI: https://doi.org/10.1007/s43069-022-00141-z