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April  2017, 13(2): 649-658. doi: 10.3934/jimo.2016038

A class of descent four–term extension of the Dai–Liao conjugate gradient method based on the scaled memoryless BFGS update

Saman Babaie–Kafaki 1,, and  Reza Ghanbari 2,

1. 

Department of Mathematics, Faculty of Mathematics, Statistics and Computer Science, Semnan University, P.O. Box: 35195–363, Semnan, Iran
2. 

Faculty of Mathematical Sciences, Ferdowsi University of Mashhad, P.O. Box: 9177948953, Mashhad, Iran

* Corresponding author: Saman Babaie–Kafaki

Received  November 2014 Revised  December 2015 Published  May 2016

Hybridizing the three–term conjugate gradient method proposed by Zhang et al. and the nonlinear conjugate gradient method proposed by Dai and Liao based on the scaled memoryless BFGS update, a one–parameter class of four–term conjugate gradient methods is proposed. It is shown that the suggested class of conjugate gradient methods possesses the sufficient descent property, without convexity assumption on the objective function. A brief global convergence analysis is made for uniformly convex objective functions. Results of numerical comparisons are reported. They demonstrate efficiency of a method of the proposed class in the sense of the Dolan–Moré performance profile.

Keywords: Unconstrained optimization, large–scale optimization, conjugate gradient method, sufficient descent property, BFGS update, global convergence.
Mathematics Subject Classification: Primary: 65K05, 90C53; Secondary: 49M37.
Citation: Saman Babaie–Kafaki, Reza Ghanbari. A class of descent four–term extension of the Dai–Liao conjugate gradient method based on the scaled memoryless BFGS update. Journal of Industrial & Management Optimization, 2017, 13 (2) : 649-658. doi: 10.3934/jimo.2016038
References:
[1]

N. Andrei, Accelerated scaled memoryless BFGS preconditioned conjugate gradient algorithm for unconstrained optimization, European J. Oper. Res., 204 (2010), 410-420.  doi: 10.1016/j.ejor.2009.11.030.  Google Scholar

[2]

S. Babaie–Kafaki, On the sufficient descent condition of the Hager–Zhang conjugate gradient methods, 4OR, 12 (2014), 285-292.  doi: 10.1007/s10288-014-0255-6.  Google Scholar

[3]

S. Babaie–Kafaki and R. Ghanbari, A descent family of Dai–Liao conjugate gradient methods, Optim. Methods Softw., 29 (2014), 583-591.  doi: 10.1080/10556788.2013.833199.  Google Scholar

[4]

S. Babaie–Kafaki and R. Ghanbari, Two modified three–term conjugate gradient methods with sufficient descent property, Optim. Lett., 8 (2014), 2285-2297.  doi: 10.1007/s11590-014-0736-8.  Google Scholar

[5]

S. Babaie–Kafaki and R. Ghanbari, A hybridization of the Hestenes–Stiefel and Dai–Yuan conjugate gradient methods based on a least–squares approach, Optim. Methods Softw., 30 (2015), 673-681.  doi: 10.1080/10556788.2014.966825.  Google Scholar

[6]

S. Babaie–Kafaki, R. Ghanbari and N. Mahdavi–Amiri, Two new conjugate gradient methods based on modified secant equations, J. Comput. Appl. Math., 234 (2010), 1374-1386.  doi: 10.1016/j.cam.2010.01.052.  Google Scholar

[7]

J. Barzilai and J. Borwein, Two–point stepsize gradient methods, IMA J. Numer. Anal., 8 (1988), 141-148.  doi: 10.1093/imanum/8.1.141.  Google Scholar

[8]

C. Broyden, The convergence of a class of double–rank minimization algorithms. Ⅱ. The new algorithm, J. Inst. Math. Appl., 6 (1970), 222-231.  doi: 10.1093/imamat/6.3.222.  Google Scholar

[9]

Y. Dai and C. Kou, A nonlinear conjugate gradient algorithm with an optimal property and an improved Wolfe line search, SIAM J. Optim., 23 (2013), 296-320.  doi: 10.1137/100813026.  Google Scholar

[10]

Y. Dai and L. Liao, New conjugacy conditions and related nonlinear conjugate gradient methods, Appl. Math. Optim., 43 (2001), 87-101.  doi: 10.1007/s002450010019.  Google Scholar

[11]

E. Dolan and J. Moré, Benchmarking optimization software with performance profiles, Math. Program., 91 (2002), 201-213.  doi: 10.1007/s101070100263.  Google Scholar

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J. Ford, Y. Narushima and H. Yabe, Multi–step nonlinear conjugate gradient methods for unconstrained minimization, Comput. Optim. Appl., 40 (2008), 191-216.  doi: 10.1007/s10589-007-9087-z.  Google Scholar

[15]

D. Goldfarb, A family of variable metric methods derived by variational means, Math. Comp., 24 (1970), 23-26.  doi: 10.1090/S0025-5718-1970-0258249-6.  Google Scholar

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N. Gould, D. Orban and P. Toint, CUTEr: a constrained and unconstrained testing environment, revisited, ACM Trans. Math. Softw., 29 (2003), 353-372.  doi: 10.1145/962437.962438.  Google Scholar

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W. Hager and H. Zhang, A new conjugate gradient method with guaranteed descent and an efficient line search, SIAM J. Optim., 16 (2005), 170-192.  doi: 10.1137/030601880.  Google Scholar

[18]

W. Hager and H. Zhang, Algorithm 851: CG_Descent, a conjugate gradient method with guaranteed descent, ACM Trans. Math. Softw., 32 (2006), 113-137.  doi: 10.1145/1132973.1132979.  Google Scholar

[19]

W. Hager and H. Zhang, A survey of nonlinear conjugate gradient methods, Pac. J. Optim., 2 (2006), 35-58.   Google Scholar

[20]

M. Hestenes and E. Stiefel, Methods of conjugate gradients for solving linear systems, J. Research Nat. Bur. Standards, 49 (1952), 409-436.  doi: 10.6028/jres.049.044.  Google Scholar

[21]

D. Li and M. Fukushima, A modified BFGS method and its global convergence in nonconvex minimization, J. Comput. Appl. Math., 129 (2001), 15-35.  doi: 10.1016/S0377-0427(00)00540-9.  Google Scholar

[22]

G. Li, C. Tang and Z. Wei, New conjugacy condition and related new conjugate gradient methods for unconstrained optimization, J. Comput. Appl. Math., 202 (2007), 523-539.  doi: 10.1016/j.cam.2006.03.005.  Google Scholar

[23]

D. Liu and J. Nocedal, On the limited memory BFGS method for large–scale optimization, Math. Program., 45 (1989), 503-528.  doi: 10.1007/BF01589116.  Google Scholar

[24]

S. Oren, Self–scaling variable metric (SSVM) algorithms. Ⅱ. Implementation and experiments, Management Sci., 20 (1974), 863-874.   Google Scholar

[25]

S. Oren and D. Luenberger, Self–scaling variable metric (SSVM) algorithms. Ⅰ. Criteria and sufficient conditions for scaling a class of algorithms, Management Sci., 20 (1973/74), 845-862.   Google Scholar

[26]

S. Oren and E. Spedicato, Optimal conditioning of self–scaling variable metric algorithms, Math. Program., 10 (1976), 70-90.  doi: 10.1007/BF01580654.  Google Scholar

[27]

D. Shanno, Conditioning of quasi–Newton methods for function minimization, Math. Comp., 24 (1970), 647-656.  doi: 10.1090/S0025-5718-1970-0274029-X.  Google Scholar

[28]

K. Sugiki, Y. Narushima and H. Yabe, Globally convergent three–term conjugate gradient methods that use secant conditions and generate descent search directions for unconstrained optimization, J. Optim. Theory Appl., 153 (2012), 733-757.  doi: 10.1007/s10957-011-9960-x.  Google Scholar

[29]

W. Sun and Y. Yuan, Optimization Theory and Methods: Nonlinear Programming, Springer, New York, 2006.  Google Scholar

[30]

Z. Wei, G. Li and L. Qi, New quasi–Newton methods for unconstrained optimization problems, Appl. Math. Comput., 175 (2006), 1156-1188.  doi: 10.1016/j.amc.2005.08.027.  Google Scholar

[31]

H. Yabe and M. Takano, Global convergence properties of nonlinear conjugate gradient methods with modified secant condition, Comput. Optim. Appl., 28 (2004), 203-225.  doi: 10.1023/B:COAP.0000026885.81997.88.  Google Scholar

[32]

Y. Yuan, A modified BFGS algorithm for unconstrained optimization, IMA J. Numer. Anal., 11 (1991), 325-332.  doi: 10.1093/imanum/11.3.325.  Google Scholar

[33]

J. Zhang, N. Deng and L. Chen, New quasi–Newton equation and related methods for unconstrained optimization, J. Optim. Theory Appl., 102 (1999), 147-167.  doi: 10.1023/A:1021898630001.  Google Scholar

[34]

L. Zhang, W. Zhou and D. Li, Some descent three–term conjugate gradient methods and their global convergence, Optim. Methods Softw., 22 (2007), 697-711.  doi: 10.1080/10556780701223293.  Google Scholar

[35]

W. Zhou and L. Zhang, A nonlinear conjugate gradient method based on the MBFGS secant condition, Optim. Methods Softw., 21 (2006), 707-714.  doi: 10.1080/10556780500137041.  Google Scholar

show all references

References:
[1]

N. Andrei, Accelerated scaled memoryless BFGS preconditioned conjugate gradient algorithm for unconstrained optimization, European J. Oper. Res., 204 (2010), 410-420.  doi: 10.1016/j.ejor.2009.11.030.  Google Scholar

[2]

S. Babaie–Kafaki, On the sufficient descent condition of the Hager–Zhang conjugate gradient methods, 4OR, 12 (2014), 285-292.  doi: 10.1007/s10288-014-0255-6.  Google Scholar

[3]

S. Babaie–Kafaki and R. Ghanbari, A descent family of Dai–Liao conjugate gradient methods, Optim. Methods Softw., 29 (2014), 583-591.  doi: 10.1080/10556788.2013.833199.  Google Scholar

[4]

S. Babaie–Kafaki and R. Ghanbari, Two modified three–term conjugate gradient methods with sufficient descent property, Optim. Lett., 8 (2014), 2285-2297.  doi: 10.1007/s11590-014-0736-8.  Google Scholar

[5]

S. Babaie–Kafaki and R. Ghanbari, A hybridization of the Hestenes–Stiefel and Dai–Yuan conjugate gradient methods based on a least–squares approach, Optim. Methods Softw., 30 (2015), 673-681.  doi: 10.1080/10556788.2014.966825.  Google Scholar

[6]

S. Babaie–Kafaki, R. Ghanbari and N. Mahdavi–Amiri, Two new conjugate gradient methods based on modified secant equations, J. Comput. Appl. Math., 234 (2010), 1374-1386.  doi: 10.1016/j.cam.2010.01.052.  Google Scholar

[7]

J. Barzilai and J. Borwein, Two–point stepsize gradient methods, IMA J. Numer. Anal., 8 (1988), 141-148.  doi: 10.1093/imanum/8.1.141.  Google Scholar

[8]

C. Broyden, The convergence of a class of double–rank minimization algorithms. Ⅱ. The new algorithm, J. Inst. Math. Appl., 6 (1970), 222-231.  doi: 10.1093/imamat/6.3.222.  Google Scholar

[9]

Y. Dai and C. Kou, A nonlinear conjugate gradient algorithm with an optimal property and an improved Wolfe line search, SIAM J. Optim., 23 (2013), 296-320.  doi: 10.1137/100813026.  Google Scholar

[10]

Y. Dai and L. Liao, New conjugacy conditions and related nonlinear conjugate gradient methods, Appl. Math. Optim., 43 (2001), 87-101.  doi: 10.1007/s002450010019.  Google Scholar

[11]

E. Dolan and J. Moré, Benchmarking optimization software with performance profiles, Math. Program., 91 (2002), 201-213.  doi: 10.1007/s101070100263.  Google Scholar

[12]

R. Fletcher, A new approach to variable metric algorithms, Comput. J., 13 (1970), 317-322.  doi: 10.1093/comjnl/13.3.317.  Google Scholar

[13]

J. Ford and I. Moghrabi, Multi–step quasi–Newton methods for optimization, J. Comput. Appl. Math., 50 (1994), 305-323.  doi: 10.1016/0377-0427(94)90309-3.  Google Scholar

[14]

J. Ford, Y. Narushima and H. Yabe, Multi–step nonlinear conjugate gradient methods for unconstrained minimization, Comput. Optim. Appl., 40 (2008), 191-216.  doi: 10.1007/s10589-007-9087-z.  Google Scholar

[15]

D. Goldfarb, A family of variable metric methods derived by variational means, Math. Comp., 24 (1970), 23-26.  doi: 10.1090/S0025-5718-1970-0258249-6.  Google Scholar

[16]

N. Gould, D. Orban and P. Toint, CUTEr: a constrained and unconstrained testing environment, revisited, ACM Trans. Math. Softw., 29 (2003), 353-372.  doi: 10.1145/962437.962438.  Google Scholar

[17]

W. Hager and H. Zhang, A new conjugate gradient method with guaranteed descent and an efficient line search, SIAM J. Optim., 16 (2005), 170-192.  doi: 10.1137/030601880.  Google Scholar

[18]

W. Hager and H. Zhang, Algorithm 851: CG_Descent, a conjugate gradient method with guaranteed descent, ACM Trans. Math. Softw., 32 (2006), 113-137.  doi: 10.1145/1132973.1132979.  Google Scholar

[19]

W. Hager and H. Zhang, A survey of nonlinear conjugate gradient methods, Pac. J. Optim., 2 (2006), 35-58.   Google Scholar

[20]

M. Hestenes and E. Stiefel, Methods of conjugate gradients for solving linear systems, J. Research Nat. Bur. Standards, 49 (1952), 409-436.  doi: 10.6028/jres.049.044.  Google Scholar

[21]

D. Li and M. Fukushima, A modified BFGS method and its global convergence in nonconvex minimization, J. Comput. Appl. Math., 129 (2001), 15-35.  doi: 10.1016/S0377-0427(00)00540-9.  Google Scholar

[22]

G. Li, C. Tang and Z. Wei, New conjugacy condition and related new conjugate gradient methods for unconstrained optimization, J. Comput. Appl. Math., 202 (2007), 523-539.  doi: 10.1016/j.cam.2006.03.005.  Google Scholar

[23]

D. Liu and J. Nocedal, On the limited memory BFGS method for large–scale optimization, Math. Program., 45 (1989), 503-528.  doi: 10.1007/BF01589116.  Google Scholar

[24]

S. Oren, Self–scaling variable metric (SSVM) algorithms. Ⅱ. Implementation and experiments, Management Sci., 20 (1974), 863-874.   Google Scholar

[25]

S. Oren and D. Luenberger, Self–scaling variable metric (SSVM) algorithms. Ⅰ. Criteria and sufficient conditions for scaling a class of algorithms, Management Sci., 20 (1973/74), 845-862.   Google Scholar

[26]

S. Oren and E. Spedicato, Optimal conditioning of self–scaling variable metric algorithms, Math. Program., 10 (1976), 70-90.  doi: 10.1007/BF01580654.  Google Scholar

[27]

D. Shanno, Conditioning of quasi–Newton methods for function minimization, Math. Comp., 24 (1970), 647-656.  doi: 10.1090/S0025-5718-1970-0274029-X.  Google Scholar

[28]

K. Sugiki, Y. Narushima and H. Yabe, Globally convergent three–term conjugate gradient methods that use secant conditions and generate descent search directions for unconstrained optimization, J. Optim. Theory Appl., 153 (2012), 733-757.  doi: 10.1007/s10957-011-9960-x.  Google Scholar

[29]

W. Sun and Y. Yuan, Optimization Theory and Methods: Nonlinear Programming, Springer, New York, 2006.  Google Scholar

[30]

Z. Wei, G. Li and L. Qi, New quasi–Newton methods for unconstrained optimization problems, Appl. Math. Comput., 175 (2006), 1156-1188.  doi: 10.1016/j.amc.2005.08.027.  Google Scholar

[31]

H. Yabe and M. Takano, Global convergence properties of nonlinear conjugate gradient methods with modified secant condition, Comput. Optim. Appl., 28 (2004), 203-225.  doi: 10.1023/B:COAP.0000026885.81997.88.  Google Scholar

[32]

Y. Yuan, A modified BFGS algorithm for unconstrained optimization, IMA J. Numer. Anal., 11 (1991), 325-332.  doi: 10.1093/imanum/11.3.325.  Google Scholar

[33]

J. Zhang, N. Deng and L. Chen, New quasi–Newton equation and related methods for unconstrained optimization, J. Optim. Theory Appl., 102 (1999), 147-167.  doi: 10.1023/A:1021898630001.  Google Scholar

[34]

L. Zhang, W. Zhou and D. Li, Some descent three–term conjugate gradient methods and their global convergence, Optim. Methods Softw., 22 (2007), 697-711.  doi: 10.1080/10556780701223293.  Google Scholar

[35]

W. Zhou and L. Zhang, A nonlinear conjugate gradient method based on the MBFGS secant condition, Optim. Methods Softw., 21 (2006), 707-714.  doi: 10.1080/10556780500137041.  Google Scholar

Figure 1.  Total number of function and gradient evaluations performance profiles for NMDL1, NMDL2, NMDL3 and MDL
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Figure 2.  CPU time performance profiles for NMDL1, NMDL2, NMDL3 and MDL
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