Abstract
In this paper, we present a new conjugate gradient method, in which the search direction is computed by minimizing a selected approximate model in a two-dimensional subspace. That is, if the objective function is not close to a quadratic, the search direction is generated by a conic model. Otherwise, a quadratic model is considered. The direction of the proposed method is proved to possess the sufficient descent property. With the modified nonmonotone line search, we establish a global convergence of the proposed method under appropriate assumptions. R-linear convergence of the proposed method is also analyzed. Numerical results using two different test function collections show that the proposed algorithm is efficient.
Similar content being viewed by others
References
Andrei N (2008) An unconstrained optimization test functions collection. Adv Model Optim 10(1):147–161
Andrei N (2014) An accelerated subspace minimization three-term conjugate gradient algorithm for unconstrained optimization. Numer Algorithms 65(4):859–874
Andrei N (2017) Accelerated adaptive Perry conjugate gradient algorithms based on the self-scaling memoryless BFGS update. J Comput Appl Math 325:149–164
Ariyawansa KA (1990) Deriving collinear scaling algorithms as extension of quasi-Newton methods and the local convergence of DFP and BFGS-related collinear scaling algorithm. Math Program 49(1):23–48
Barzilai J, Borwein JM (1988) Two-point step size gradient methods. IMA J Numer Anal 8:141–148
Branch MA, Coleman TF, Li Y (1999) A subspace, interior, and conjugate gradient method for large scale bound-constrained minimization problems. SIAM J Sci Comput 21(1):1–23
Dai YH, Kou CX (2013) A nonlinear conjugate gradient algorithm with an optimal property and an improved Wolfe line search. SIAM J Optim 23(1):296–320
Dai YH, Kou CX (2016) A Barzilai–Borwein conjugate gradient method. Sci China Math 59(8):1511–1524
Dai YH, Liao LZ (2001) New conjugacy conditions and related nonlinear conjugate gradient methods. Appl Math Optim 43(1):87–101
Dai YH, Yuan Y (1999) A nonlinear conjugate gradient method with a strong global convergence property. SIAM J Optim 10(1):177–182
Dai YH, Yuan YX (2000) Nonlinear conjugate gradient methods. Shanghai Scientific and Technical Publishers, Shanghai
Davidon WC (1980) Conic approximations and collinear scalings for optimizers. SIAM J Numer Anal 17(2):268–281
Di S, Sun WY (1996) A trust region method for conic model to solve unconstrained optimization. Optim Methods Softw 6(4):237–263
Dolan ED, Moré JJ (2002) Benchmarking optimization software with performance profiles. Math Program 91:201–213
Du XW, Zhang P, Ma WY (2016) Some modified conjugate gradient methods for unconstrained optimization. J Comput Appl Math 305:92–114
Facchinei F, Pang JS (2003) Finite-dimensional variational inequalities and complementarity problems, vol 2. Springer, Berlin
Fletcher R, Reeves CM (1964) Function minimization by conjugate gradients. Comput J 7:149–154
Gould NIM, Orban D, Toint PhL (2003) CUTEr and SifDec: a constrained and unconstrained testing environment, revisited. ACM Trans Math Softw 29(4):373–394
Hager WW, Zhang H (2005) A new conjugate gradient method with guaranteed descent and an efficient line search. SIAM J Optim 16(1):170–192
Hager WW, Zhang H (2006a) A survey of nonlinear conjugate gradient methods. Pac J Optim 2(1):35–58
Hager WW, Zhang H (2006b) Algorithm 851: CG_DESCENT, a conjugate gradient method with guaranteed descent. ACM Trans Math Softw 32(1):113–137
Hager WW, Zhang H (2013) The limited memory conjugate gradient method. SIAM J Optim 23(4):2150–2168
Hestenes MR, Stiefel E (1952) Methods of conjugate gradients for solving linear systems. J Res Natl Bur Stand 49:409–436
Li M, Liu HW, Liu ZX (2018) A new subspace minimization conjugate gradient method with nonmonotone line search for unconstrained optimization. Numer Algorithms 79(1):195–219
Liu HW, Liu ZX (2018a) An efficient Barzilai–Borwein conjugate gradient method for unconstrained optimization. J Optim Theory Appl. https://doi.org/10.1007/s10957-018-1393-3
Liu ZX, Liu HW (2018b) An efficient gradient method with approximate optimal stepsize for large-scale unconstrained optimization. Numer. Algorithms 78(1):21–39
Liu ZX, Liu HW (2018c) Several efficient gradient methods with approximate optimal stepsizes for large scale unconstrained optimization. J Comput Appl Math 328:400–413
Nocedal J, Wright SJ (2006) Numerical optimization, 2nd edn. Springer, Berlin
Polak E, Ribière G (1969) Note sur la convergence de méthodes de directions conjuguées. Rev. Franaise Informat. Rech. Opérationnelle. 3(16):35–43
Polyak BT (1969) The conjugate gradient method in extremal problems. Ussr Comput Math Math Phys 9(4):94–112
Raydan M (1997) The Barzilai and Borwein gradient method for the large scale unconstrained minimization problem. SIAM J Optim 7(1):26–33
Schnabel RB (1982) Conic methods for unconstrained minimization and tensor methods for nonlinear equation. In: Mathamatical Program, pp 417–438
Sheng S (1995) Interpolation by conic model for unconstrained optimization. Computing 54(1):83–98
Sorensen DC (1980) The Q-Superlinear convergence of a collinear scaling algorithm for unconstrained optimization. SIAM J Numer Anal 17(1):84–114
Sun WY (1996) On nonquadratic model optimization methods. Asia Pac J Oper Res 13:43–63
Sun WY, Yuan YX (2001) A conic trust-region method for nonlinearly constrained optimization. Ann. Oper. Res. 103:175–191
Sun W, Yuan J, Yuan Y (2003) Trust region method of conic model for linearly constrained optimization. J Comput Math 21:295–304
Yang YT, Chen YT, Lu YL (2017) A subspace conjugate gradient algorithm for large-scale unconstrained optimization. Numer Algorithms 76(3):813–828
Yuan YX (1991) A modified BFGS algorithm for unconstrained optimization. IMA J Numer Anal 11(3):325–332
Yuan YX (2009) Subspace methods for large scale nonlinear equations and nonlinear least squares. Optim Eng 10(2):207–218
Yuan YX (2014) A review on subspace methods for nonlinear optimization. In: Proceedings of the international congress of mathematics. Korea, pp 807–827
Yuan YX, Stoer J (1995) A subspace study on conjugate gradient algorithms. Z Angew Math Mech 75(1):69–77
Yuan YX, Sun WY (1997) Optimization theory and methods. Science Press, Beijing
Zhang H, Hager WW (2004) A nonmonotone line search technique and its application to unconstrained optimization. SIAM J Optim 14(4):1043–1056
Acknowledgements
We would like to thank the anonymous referees for their useful suggestions and comments. We also would like to thank Professor Dai, Y. H. and Dr. Kou, C. X. for their CGOPT code, and thank Professor Hager, W. W. and Zhang, H. C. for their CG DESCENT (5.3) code. This research is supported by National Science Foundation of China (No. 11461021) and Shaanxi Science Foundation (No. 2017JM1014).
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Hector Ramirez.
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
Li, Y., Liu, Z. & Liu, H. A subspace minimization conjugate gradient method based on conic model for unconstrained optimization. Comp. Appl. Math. 38, 16 (2019). https://doi.org/10.1007/s40314-019-0779-7
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s40314-019-0779-7