Hybridizing self-adjusting approach of Dong et al. and three-term formulation of Zhang et al., a nonlinear conjugate gradient method is proposed. The method reduces to the Polak–Ribière–Polyak method under the exact line search and satisfies the sufficient descent condition independent of the line search and the objective function convexity. Similar to the Polak–Ribière–Polyak method, the method possesses an automatic restart feature which avoids jamming. Global convergence analyses are conducted when the line search fulfills the popular Wolfe conditions as well as an Armijo-type condition. Numerical experiments are done on a set of CUTEr unconstrained optimization test problems. Results of comparisons show computational efficiency of the proposed method in the sense of Dolan–Moré performance profile.

Accepted:

DOI: 10.1051/ro/2016009

Keywords: Unconstrained optimization, conjugate gradient method, sufficient descent condition, line search, global convergence

^{1}; Ghanbari, Reza

^{2}

@article{RO_2016__50_3_567_0, author = {Babaie-Kafaki, Saman and Ghanbari, Reza}, title = {A descent hybrid modification of the {Polak{\textendash}Ribi\`ere{\textendash}Polyak} conjugate gradient method}, journal = {RAIRO - Operations Research - Recherche Op\'erationnelle}, pages = {567--574}, publisher = {EDP-Sciences}, volume = {50}, number = {3}, year = {2016}, doi = {10.1051/ro/2016009}, mrnumber = {3538840}, zbl = {1354.90174}, language = {en}, url = {http://archive.numdam.org/articles/10.1051/ro/2016009/} }

TY - JOUR AU - Babaie-Kafaki, Saman AU - Ghanbari, Reza TI - A descent hybrid modification of the Polak–Ribière–Polyak conjugate gradient method JO - RAIRO - Operations Research - Recherche Opérationnelle PY - 2016 SP - 567 EP - 574 VL - 50 IS - 3 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/ro/2016009/ DO - 10.1051/ro/2016009 LA - en ID - RO_2016__50_3_567_0 ER -

%0 Journal Article %A Babaie-Kafaki, Saman %A Ghanbari, Reza %T A descent hybrid modification of the Polak–Ribière–Polyak conjugate gradient method %J RAIRO - Operations Research - Recherche Opérationnelle %D 2016 %P 567-574 %V 50 %N 3 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/ro/2016009/ %R 10.1051/ro/2016009 %G en %F RO_2016__50_3_567_0

Babaie-Kafaki, Saman; Ghanbari, Reza. A descent hybrid modification of the Polak–Ribière–Polyak conjugate gradient method. RAIRO - Operations Research - Recherche Opérationnelle, Volume 50 (2016) no. 3, pp. 567-574. doi : 10.1051/ro/2016009. http://archive.numdam.org/articles/10.1051/ro/2016009/

A modified Polak–Ribière–Polyak conjugate gradient algorithm for unconstrained optimization. Optimization 60 (2011) 1457–1471. | DOI | MR | Zbl

,An adaptive conjugate gradient algorithm for large–scale unconstrained optimization. J. Comput. Appl. Math. 292 (2016) 83–91. | DOI | MR | Zbl

,An eigenvalue study on the sufficient descent property of a modified Polak–Ribière–Polyak conjugate gradient method. Bull. Iranian Math. Soc. 40 (2014) 235–242. | MR | Zbl

,A descent extension of the Polak–Ribière–Polyak conjugate gradient method. Comput. Math. Appl. 68 (2014) 2005–2011. | DOI | MR | Zbl

and ,A two-term PRP-based descent method. Numer. Funct. Anal. Optim. 28 (2007) 1217–1230. | DOI | MR | Zbl

,New conjugacy conditions and related nonlinear conjugate gradient methods. Appl. Math. Optim. 43 (2001) 87–101. | DOI | MR | Zbl

and ,Convergence properties of nonlinear conjugate gradient methods. SIAM J. Optim. 10 (1999) 348–358. | Zbl

, , , , and ,An improved spectral conjugate gradient algorithm for nonconvex unconstrained optimization problems. J. Optim. Theory Appl. 157 (2013) 820–842. | DOI | MR | Zbl

, and ,Benchmarking optimization software with performance profiles. Math. Program. 91 (2002) 201–213. | DOI | MR | Zbl

and ,New version of the three-term conjugate gradient method based on spectral scaling conjugacy condition that generates descent search direction. Appl. Math. Comput. 269 (2015) 606–617. | DOI | MR | Zbl

, and ,A self-adjusting conjugate gradient method with sufficient descent condition and conjugacy condition. J. Optim. Theory Appl. 165 (2015) 225–241. | DOI | MR | Zbl

, and ,A modified Hestenes–Stiefel conjugate gradient method with sufficient descent condition and conjugacy condition. J. Comput. Appl. Math. 281 (2015) 239–249. | DOI | MR | Zbl

, , and ,CUTEr: a constrained and unconstrained testing environment, revisited. ACM Trans. Math. Softw. 29 (2003) 373–394. | DOI | Zbl

, and ,Algorithm 851: CG−Descent, a conjugate gradient method with guaranteed descent. ACM Trans. Math. Softw. 32 (2006) 113–137. | DOI | MR | Zbl

and ,A survey of nonlinear conjugate gradient methods. Pac. J. Optim. 2 (2006) 35–58. | MR | Zbl

and ,Methods of conjugate gradients for solving linear systems. J. Res. Nat. Bur. Standards 49 (1952) 409–436. | DOI | MR | Zbl

and ,Note sur la convergence de méthodes de directions conjuguées. Rev. Française Informat. Recherche Opérationnelle 3 (1969) 35–43. | Numdam | MR | Zbl

and ,The conjugate gradient method in extreme problems. USSR Comp. Math. Math. Phys. 9 (1969) 94–112. | DOI | Zbl

,W. Sun and Y.X. Yuan, Optimization Theory and Methods: Nonlinear Programming. Springer, New York (2006). | MR | Zbl

New spectral PRP conjugate gradient method for unconstrained optimization. Appl. Math. Lett. 24 (2011) 16–22. | DOI | MR | Zbl

, and ,Global convergence of modified Polak–Ribière–Polyak conjugate gradient methods with sufficient descent property. J. Ind. Manag. Optim. 4 (2008) 565–579. | DOI | MR | Zbl

, and ,Modified nonlinear conjugate gradient methods with sufficient descent property for large-scale optimization problems. Optim. Lett. 3 (2009) 11–21. | DOI | MR | Zbl

,A descent modified Polak–Ribière–Polyak conjugate gradient method and its global convergence. IMA J. Numer. Anal. 26 (2006) 629–640. | DOI | MR | Zbl

, and ,*Cited by Sources: *