September 2018, 8(3): 377-387. doi: 10.3934/naco.2018024

Multi-step spectral gradient methods with modified weak secant relation for large scale unconstrained optimization

1. 

Department of Mathematical and Actuarial Sciences, Lee Kong Chian Faculty of Engineering and Science, Universiti Tunku Abdul Rahman, Sungai Long Campus, Jalan Sungai Long 9, Bandar Sungai Long, 43000 Kajang, Selangor, Malaysia

2. 

Department of Mathematics, Faculty of Science, Universiti Putra Malaysia, 43400 Serdang, Selangor, Malaysia

3. 

Institute for Mathematical Research, Universiti Putra Malaysia, 43400 Serdang, Selangor, Malaysia

* Corresponding author: Hong Seng Sim

Received  May 2017 Revised  March 2018 Published  June 2018

Fund Project: The first author is supported by Yayasan Sultan Iskandar Johor 2014

In this paper, we aim to propose some spectral gradient methods via variational technique under log-determinant norm. The spectral parameters satisfy the modified weak secant relations that inspired by the multistep approximation for solving large scale unconstrained optimization. An executable code is developed to test the efficiency of the proposed method with spectral gradient method using standard weak secant relation as constraint. Numerical results are presented which suggest a better performance has been achieved.

Citation: Hong Seng Sim, Wah June Leong, Chuei Yee Chen, Siti Nur Iqmal Ibrahim. Multi-step spectral gradient methods with modified weak secant relation for large scale unconstrained optimization. Numerical Algebra, Control & Optimization, 2018, 8 (3) : 377-387. doi: 10.3934/naco.2018024
References:
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N. Andrei, An unconstrained optimization test functions collection, Advanced Modeling and Optimization, 10 (2008), 147-161.

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J. Barzilai and J. Borwein, Two-point step size gradient methods, IMA Journal of Numerical Analysis, 8 (1988), 141-148. doi: 10.1093/imanum/8.1.141.

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R. H. Byrd and J. Nocedal, A tool for the analysis of quasi-Newton methods with application to unconstrained minimization, SIAM J. Numer. Anal., 26 (1989), 727-739. doi: 10.1137/0726042.

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J. E. Dennis and H. Wolkowicz, Sizing and least change secant methods, SIAM Journal on Numerical Analysis, 30 (1993), 1291-1313. doi: 10.1137/0730067.

[6]

J. A. Ford and I. A. Moghrabi, Multi-step quasi-Newton methods for optimization, Journal of Computational and Applied Mathematics, 50 (1994), 305-323. doi: 10.1016/0377-0427(94)90309-3.

[7]

W. La CruzJ. Martinez and M. Raydan, Spectral residual method without gradient information for solving large-scale nonlinear systems of equations, Math. Comput., 75 (2006), 1449-1466. doi: 10.1090/S0025-5718-06-01840-0.

[8]

W. La Cruz and M. Raydan, Nonmonotone spectral methods for large-scale nonlinear systems, Optim. Methods Softw., 18 (2003), 583-599. doi: 10.1080/10556780310001610493.

[9]

D. C. Liu and J. Nocedal, On the limited memory BFGS method for large scale optimization, Mathematical Programming, 45 (1989), 503-528. doi: 10.1007/BF01589116.

[10]

M. Raydan, The Barzilai and Borwein gradient method for the large scale unconstrained minimization problem, SIAM Journal of Optimization, 7 (1997), 26-33. doi: 10.1137/S1052623494266365.

[11]

M. ZhuJ. L. Nazareth and H. Wolkowicz, The quasi-Cauchy relation and diagonal updating, SIAM Journal on Optimization, 4 (1999), 1192-1204. doi: 10.1137/S1052623498331793.

show all references

References:
[1]

N. Andrei, An unconstrained optimization test functions collection, Advanced Modeling and Optimization, 10 (2008), 147-161.

[2]

J. Barzilai and J. Borwein, Two-point step size gradient methods, IMA Journal of Numerical Analysis, 8 (1988), 141-148. doi: 10.1093/imanum/8.1.141.

[3]

I. BongartzA. R. ConnN. I. M. Gould and Ph. L. Toint, CUTE: constrained and unconstrained testing environment, ACM Transactions on Mathematical Software, 21 (1995), 123-160. doi: 10.1145/200979.201043.

[4]

R. H. Byrd and J. Nocedal, A tool for the analysis of quasi-Newton methods with application to unconstrained minimization, SIAM J. Numer. Anal., 26 (1989), 727-739. doi: 10.1137/0726042.

[5]

J. E. Dennis and H. Wolkowicz, Sizing and least change secant methods, SIAM Journal on Numerical Analysis, 30 (1993), 1291-1313. doi: 10.1137/0730067.

[6]

J. A. Ford and I. A. Moghrabi, Multi-step quasi-Newton methods for optimization, Journal of Computational and Applied Mathematics, 50 (1994), 305-323. doi: 10.1016/0377-0427(94)90309-3.

[7]

W. La CruzJ. Martinez and M. Raydan, Spectral residual method without gradient information for solving large-scale nonlinear systems of equations, Math. Comput., 75 (2006), 1449-1466. doi: 10.1090/S0025-5718-06-01840-0.

[8]

W. La Cruz and M. Raydan, Nonmonotone spectral methods for large-scale nonlinear systems, Optim. Methods Softw., 18 (2003), 583-599. doi: 10.1080/10556780310001610493.

[9]

D. C. Liu and J. Nocedal, On the limited memory BFGS method for large scale optimization, Mathematical Programming, 45 (1989), 503-528. doi: 10.1007/BF01589116.

[10]

M. Raydan, The Barzilai and Borwein gradient method for the large scale unconstrained minimization problem, SIAM Journal of Optimization, 7 (1997), 26-33. doi: 10.1137/S1052623494266365.

[11]

M. ZhuJ. L. Nazareth and H. Wolkowicz, The quasi-Cauchy relation and diagonal updating, SIAM Journal on Optimization, 4 (1999), 1192-1204. doi: 10.1137/S1052623498331793.

Figure 1.  Performance Profiling for the Modified Multiple Spectral Gradient Methods and Standard Multiple Spectral Gradient Method in terms of Number of Iterations.
Figure 2.  Performance Profiling for the Modified Multiple Spectral Gradient Methods and Standard Multiple Spectral Gradient Method in terms of Number of Function Calls.
Figure 3.  Performance Profiling for the Modified Multiple Spectral Gradient Methods and Standard Multiple Spectral Gradient Method in terms of CPU Time.
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