doi: 10.3934/naco.2019029

Numerical comparisons of smoothing functions for optimal correction of an infeasible system of absolute value equations

Department of Applied Mathematics, Faculty of Mathematical Sciences, University of Guilan, Rasht, Iran

* Corresponding author: Saeed Ketabchi

Received  June 2018 Revised  April 2019 Published  May 2019

Optimal correction of an infeasible system of absolute value equations (AVEs), leads into a nonconvex and nonsmooth fractional problem. Using Dinkelbach's approach, this problem can be reformulated to form a single variable equation. In this paper, first, we have smoothed the equation by considering four important and famous smoothing functions (see[2,23]) and thus, to solve it, a smoothing-type algorithm based on the Difference of Convex (DC) algorithm-Newtown methods is proposed. Finally, the randomly generated AVEs were compared to find the best smoothing function.

Citation: Fakhrodin Hashemi, Saeed Ketabchi. Numerical comparisons of smoothing functions for optimal correction of an infeasible system of absolute value equations. Numerical Algebra, Control & Optimization, doi: 10.3934/naco.2019029
References:
[1]

F. J. A. ArtachoR. M. Fleming and P. T. Vuong, Accelerating the dc algorithm for smooth functions, Mathematical Programming, 169 (2018), 95-118.  doi: 10.1007/s10107-017-1180-1.  Google Scholar

[2]

X. Chen, Smoothing methods for nonsmooth, nonconvex minimization, Mathematical Programming, 134 (2012), 71-99.  doi: 10.1007/s10107-012-0569-0.  Google Scholar

[3]

F. H. Clarke, Optimization and Nonsmooth Analysis, Vol. 5, SIAM, 1990. doi: 10.1137/1.9781611971309.  Google Scholar

[4]

W. Dinkelbach, On nonlinear fractional programming, Management Science, 13 (1967), 492-498.  doi: 10.1287/mnsc.13.7.492.  Google Scholar

[5]

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

[6]

S.-L. Hu and Z.-H. Huang, A note on absolute value equations, Optimization Letters, 4 (2010), 417-424.  doi: 10.1007/s11590-009-0169-y.  Google Scholar

[7]

X. Jiang and Y. Zhang, A smoothing-type algorithm for absolute value equations, Journal of Industrial and Management Optimization, 9 (2013), 789-798.  doi: 10.3934/jimo.2013.9.789.  Google Scholar

[8]

S. Ketabchi and H. Moosaei, An efficient method for optimal correcting of absolute value equations by minimal changes in the right hand side, Computers and Mathematics with Applications, 64 (2012), 1882-1885.  doi: 10.1016/j.camwa.2012.03.015.  Google Scholar

[9]

S. Ketabchi and H. Moosaei, Minimum norm solution to the absolute value equation in the convex case, Journal of Optimization Theory and Applications, 154 (2012), 1080-1087.  doi: 10.1007/s10957-012-0044-3.  Google Scholar

[10]

S. Ketabchi and H. Moosaei, Optimal error correction and methods of feasible directions, Journal of Optimization Theory and Applications, 154 (2012), 209-216.  doi: 10.1007/s10957-012-0009-6.  Google Scholar

[11]

S. KetabchiH. Moosaei and S. Fallahi, Optimal error correction of the absolute value equation using a genetic algorithm, Mathematical and Computer Modelling, 57 (2013), 2339-2342.   Google Scholar

[12]

O. L. Mangasarian, Absolute value equation solution via concave minimization, Optimization Letters, 1 (2007), 3-8.  doi: 10.1007/s11590-006-0005-6.  Google Scholar

[13]

O. L. Mangasarian, Absolute value programming, Computational Optimization and Applications, 36 (2007), 43-53.  doi: 10.1007/s10589-006-0395-5.  Google Scholar

[14]

O. L. Mangasarian, A generalized newton method for absolute value equations, Optimization Letters, 3 (2009), 101-108.  doi: 10.1007/s11590-008-0094-5.  Google Scholar

[15]

O. L. Mangasarian, Primal-dual bilinear programming solution of the absolute value equation, Optimization Letters, 6 (2012), 1527-1533.  doi: 10.1007/s11590-011-0347-6.  Google Scholar

[16]

O. L. Mangasarian, Absolute value equation solution via dual complementarity, Optimization Letters, 7 (2013), 625-630.  doi: 10.1007/s11590-012-0469-5.  Google Scholar

[17]

O. L. Mangasarian and R. R. Meyer, Absolute value equations, Linear Algebra and Its Applications, 419 (2006), 359-367.  doi: 10.1016/j.laa.2006.05.004.  Google Scholar

[18]

H. MoosaeiS. Ketabchi and P. M. Pardalos, Tikhonov regularization for infeasible absolute value equations, Optimization, 65 (2016), 1531-1537.  doi: 10.1080/02331934.2016.1154963.  Google Scholar

[19]

A. Nekvinda and L. Zajíček, A simple proof of the rademacher theorem, Časopis pro pěstování matematiky, 113 (1988), 337–341.  Google Scholar

[20]

J. Nocedal and S. J. Wright, Numerical Optimization, 2nd edition, Springer, 2006.  Google Scholar

[21]

O. Prokopyev, On equivalent reformulations for absolute value equations, Computational Optimization and Applications, 44 (2009), 363-372.  doi: 10.1007/s10589-007-9158-1.  Google Scholar

[22]

J. Rohn, A theorem of the alternatives for the equation A|x|+ B|x|=b, Linear and Multilinear Algebra, 52 (2004), 421-426.  doi: 10.1080/0308108042000220686.  Google Scholar

[23]

B. SaheyaC.-H. Yu and J.-S. Chen, Numerical comparisons based on four smoothing functions for absolute value equation, Journal of Applied Mathematics and Computing, 56 (2018), 131-149.  doi: 10.1007/s12190-016-1065-0.  Google Scholar

[24]

P. D. Tao and L. T. H. An, Convex analysis approach to dc programming: Theory, algorithms and applications, Acta Mathematica Vietnamica, 22 (1997), 289-355.   Google Scholar

show all references

References:
[1]

F. J. A. ArtachoR. M. Fleming and P. T. Vuong, Accelerating the dc algorithm for smooth functions, Mathematical Programming, 169 (2018), 95-118.  doi: 10.1007/s10107-017-1180-1.  Google Scholar

[2]

X. Chen, Smoothing methods for nonsmooth, nonconvex minimization, Mathematical Programming, 134 (2012), 71-99.  doi: 10.1007/s10107-012-0569-0.  Google Scholar

[3]

F. H. Clarke, Optimization and Nonsmooth Analysis, Vol. 5, SIAM, 1990. doi: 10.1137/1.9781611971309.  Google Scholar

[4]

W. Dinkelbach, On nonlinear fractional programming, Management Science, 13 (1967), 492-498.  doi: 10.1287/mnsc.13.7.492.  Google Scholar

[5]

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

[6]

S.-L. Hu and Z.-H. Huang, A note on absolute value equations, Optimization Letters, 4 (2010), 417-424.  doi: 10.1007/s11590-009-0169-y.  Google Scholar

[7]

X. Jiang and Y. Zhang, A smoothing-type algorithm for absolute value equations, Journal of Industrial and Management Optimization, 9 (2013), 789-798.  doi: 10.3934/jimo.2013.9.789.  Google Scholar

[8]

S. Ketabchi and H. Moosaei, An efficient method for optimal correcting of absolute value equations by minimal changes in the right hand side, Computers and Mathematics with Applications, 64 (2012), 1882-1885.  doi: 10.1016/j.camwa.2012.03.015.  Google Scholar

[9]

S. Ketabchi and H. Moosaei, Minimum norm solution to the absolute value equation in the convex case, Journal of Optimization Theory and Applications, 154 (2012), 1080-1087.  doi: 10.1007/s10957-012-0044-3.  Google Scholar

[10]

S. Ketabchi and H. Moosaei, Optimal error correction and methods of feasible directions, Journal of Optimization Theory and Applications, 154 (2012), 209-216.  doi: 10.1007/s10957-012-0009-6.  Google Scholar

[11]

S. KetabchiH. Moosaei and S. Fallahi, Optimal error correction of the absolute value equation using a genetic algorithm, Mathematical and Computer Modelling, 57 (2013), 2339-2342.   Google Scholar

[12]

O. L. Mangasarian, Absolute value equation solution via concave minimization, Optimization Letters, 1 (2007), 3-8.  doi: 10.1007/s11590-006-0005-6.  Google Scholar

[13]

O. L. Mangasarian, Absolute value programming, Computational Optimization and Applications, 36 (2007), 43-53.  doi: 10.1007/s10589-006-0395-5.  Google Scholar

[14]

O. L. Mangasarian, A generalized newton method for absolute value equations, Optimization Letters, 3 (2009), 101-108.  doi: 10.1007/s11590-008-0094-5.  Google Scholar

[15]

O. L. Mangasarian, Primal-dual bilinear programming solution of the absolute value equation, Optimization Letters, 6 (2012), 1527-1533.  doi: 10.1007/s11590-011-0347-6.  Google Scholar

[16]

O. L. Mangasarian, Absolute value equation solution via dual complementarity, Optimization Letters, 7 (2013), 625-630.  doi: 10.1007/s11590-012-0469-5.  Google Scholar

[17]

O. L. Mangasarian and R. R. Meyer, Absolute value equations, Linear Algebra and Its Applications, 419 (2006), 359-367.  doi: 10.1016/j.laa.2006.05.004.  Google Scholar

[18]

H. MoosaeiS. Ketabchi and P. M. Pardalos, Tikhonov regularization for infeasible absolute value equations, Optimization, 65 (2016), 1531-1537.  doi: 10.1080/02331934.2016.1154963.  Google Scholar

[19]

A. Nekvinda and L. Zajíček, A simple proof of the rademacher theorem, Časopis pro pěstování matematiky, 113 (1988), 337–341.  Google Scholar

[20]

J. Nocedal and S. J. Wright, Numerical Optimization, 2nd edition, Springer, 2006.  Google Scholar

[21]

O. Prokopyev, On equivalent reformulations for absolute value equations, Computational Optimization and Applications, 44 (2009), 363-372.  doi: 10.1007/s10589-007-9158-1.  Google Scholar

[22]

J. Rohn, A theorem of the alternatives for the equation A|x|+ B|x|=b, Linear and Multilinear Algebra, 52 (2004), 421-426.  doi: 10.1080/0308108042000220686.  Google Scholar

[23]

B. SaheyaC.-H. Yu and J.-S. Chen, Numerical comparisons based on four smoothing functions for absolute value equation, Journal of Applied Mathematics and Computing, 56 (2018), 131-149.  doi: 10.1007/s12190-016-1065-0.  Google Scholar

[24]

P. D. Tao and L. T. H. An, Convex analysis approach to dc programming: Theory, algorithms and applications, Acta Mathematica Vietnamica, 22 (1997), 289-355.   Google Scholar

Figure 1.  Performance profile of computing time of Algorithm 2 for the example 1
Figure 2.  Performance profile of computing time of Algorithm 2 for the example 2
Table 1.  Numerical results for Example 1
$ \phi_1 $ $ \phi_2 $ $ \phi_3 $ $ \phi_4 $
$ n $ $ Me $ $ Merror $ $ Mtime $ $ Me $ $ Merror $ $ Mtime $ $ Me $ $ Merror $ $ Mtime $ $ Me $ $ Merror $ $ Mtime $
100 8.27e-17 5.52e-11 0.04 9.84e-17 1.20e-10 0.03 8.29e-17 1.16e-10 0.04 7.88e-17 1.95e-11 0.01
500 8.42e-17 1.21e-10 1.01 8.32e-17 3.04e-10 0.83 1.03e-16 3.52e-10 0.98 8.01e-17 1.35e-10 0.38
1000 8.25e-17 1.41e-10 5.37 9.34e-17 6.02e-10 4.30 1.05e-16 4.17e-10 4.83 8.78e-17 9.15e-11 2.13
1500 9.36e-17 2.01e-10 14.07 7.98e-17 7.67e-10 11.82 1.00e-16 3.98e-10 12.13 8.80e-17 2.24e-11 6.06
2000 8.99e-17 2.68e-10 30.14 9.17e-17 4.94e-10 23.72 9.95e-17 4.78e-10 26.56 9.30e-17 6.71e-11 11.48
2500 9.42e-17 3.10e-10 48.76 7.83e-17 5.78e-10 37.08 9.00e-17 7.10e-10 46.51 9.65e-17 1.18e-10 19.34
3000 8.84e-17 3.79e-10 85.25 9.87e-17 6.94e-10 66.18 8.06e-17 6.69e-10 78.62 9.62e-17 2.07e-10 35.11
3500 8.44e-17 4.18e-10 122.58 9.77e-17 8.11e-10 98.50 9.37e-17 7.81e-10 117.32 9.43e-17 2.82e-11 52.73
4000 8.89e-17 4.67e-10 162.58 9.82e-17 9.08e-10 134.06 9.35e-17 8.94e-10 158.81 8.23e-17 4.70e-11 73.29
4500 1.01e-16 5.23e-10 245.94 9.12e-17 9.87e-10 189.79 9.14e-17 1.02e-09 242.38 8.81e-17 6.53e-11 105.15
5000 9.17e-17 5.58e-10 289.93 8.92e-17 1.09e-09 241.79 9.31e-17 1.09e-09 288.33 9.07e-17 8.83e-11 122.53
$ \phi_1 $ $ \phi_2 $ $ \phi_3 $ $ \phi_4 $
$ n $ $ Me $ $ Merror $ $ Mtime $ $ Me $ $ Merror $ $ Mtime $ $ Me $ $ Merror $ $ Mtime $ $ Me $ $ Merror $ $ Mtime $
100 8.27e-17 5.52e-11 0.04 9.84e-17 1.20e-10 0.03 8.29e-17 1.16e-10 0.04 7.88e-17 1.95e-11 0.01
500 8.42e-17 1.21e-10 1.01 8.32e-17 3.04e-10 0.83 1.03e-16 3.52e-10 0.98 8.01e-17 1.35e-10 0.38
1000 8.25e-17 1.41e-10 5.37 9.34e-17 6.02e-10 4.30 1.05e-16 4.17e-10 4.83 8.78e-17 9.15e-11 2.13
1500 9.36e-17 2.01e-10 14.07 7.98e-17 7.67e-10 11.82 1.00e-16 3.98e-10 12.13 8.80e-17 2.24e-11 6.06
2000 8.99e-17 2.68e-10 30.14 9.17e-17 4.94e-10 23.72 9.95e-17 4.78e-10 26.56 9.30e-17 6.71e-11 11.48
2500 9.42e-17 3.10e-10 48.76 7.83e-17 5.78e-10 37.08 9.00e-17 7.10e-10 46.51 9.65e-17 1.18e-10 19.34
3000 8.84e-17 3.79e-10 85.25 9.87e-17 6.94e-10 66.18 8.06e-17 6.69e-10 78.62 9.62e-17 2.07e-10 35.11
3500 8.44e-17 4.18e-10 122.58 9.77e-17 8.11e-10 98.50 9.37e-17 7.81e-10 117.32 9.43e-17 2.82e-11 52.73
4000 8.89e-17 4.67e-10 162.58 9.82e-17 9.08e-10 134.06 9.35e-17 8.94e-10 158.81 8.23e-17 4.70e-11 73.29
4500 1.01e-16 5.23e-10 245.94 9.12e-17 9.87e-10 189.79 9.14e-17 1.02e-09 242.38 8.81e-17 6.53e-11 105.15
5000 9.17e-17 5.58e-10 289.93 8.92e-17 1.09e-09 241.79 9.31e-17 1.09e-09 288.33 9.07e-17 8.83e-11 122.53
Table 2.  Numerical results for Example 2
$ \phi_1 $ $ \phi_2 $ $ \phi_3 $ $ \phi_4 $
$ n $ $ Me $ $ Merror $ $ Mtime $ $ Me $ $ Merror $ $ Mtime $ $ Me $ $ Merror $ $ Mtime $ $ Me $ $ Merror $ $ Mtime $
100 8.85e-17 2.71e-11 0.03 1.04e-16 2.09e-11 0.03 9.03e-17 5.04e-11 0.03 9.36e-17 3.17e-11 0.02
500 8.30e-17 2.55e-11 1.06 8.38e-17 1.51e-11 1.01 8.70e-17 9.62e-11 1.05 8.91e-17 3.69e-10 0.48
1000 9.17e-17 1.63e-11 6.77 8.96e-17 5.38e-12 6.42 8.11e-17 6.93e-11 6.71 8.58e-17 2.82e-10 2.63
1500 8.66e-17 1.75e-11 18.49 8.76e-17 7.15e-12 18.40 8.94e-17 3.95e-11 18.49 8.76e-17 7.39e-11 7.69
2000 8.91e-17 1.52e-11 34.22 8.68e-17 6.17e-12 32.40 9.38e-17 4.12e-11 33.71 8.62e-17 2.07e-10 17.75
2500 9.45e-17 2.82e-11 63.48 9.18e-17 3.01e-12 60.78 7.82e-17 5.16e-11 62.92 8.66e-17 3.56e-10 33.42
3000 8.06e-17 1.70e-11 113.18 8.08e-17 8.29e-12 100.24 9.50e-17 3.59e-11 112.44 9.35e-17 6.29e-10 52.83
3500 8.56e-17 6.96e-12 169.34 1.01e-16 3.62e-12 162.04 1.02e-16 2.63e-11 162.79 7.56e-17 8.45e-11 81.62
4000 8.21e-17 1.34e-11 245.96 9.40e-17 3.91e-12 244.70 8.69e-17 3.41e-11 245.37 9.57e-17 1.38e-10 126.02
4500 7.84e-17 1.70e-11 351.13 9.31e-17 4.52e-12 313.81 9.19e-17 4.93e-11 328.67 8.42e-17 2.02e-10 175.69
5000 1.11e-16 3.03e-11 476.20 8.51e-17 4.20e-12 437.35 1.03e-16 5.77e-11 444.11 6.94e-17 2.67e-10 232.96
$ \phi_1 $ $ \phi_2 $ $ \phi_3 $ $ \phi_4 $
$ n $ $ Me $ $ Merror $ $ Mtime $ $ Me $ $ Merror $ $ Mtime $ $ Me $ $ Merror $ $ Mtime $ $ Me $ $ Merror $ $ Mtime $
100 8.85e-17 2.71e-11 0.03 1.04e-16 2.09e-11 0.03 9.03e-17 5.04e-11 0.03 9.36e-17 3.17e-11 0.02
500 8.30e-17 2.55e-11 1.06 8.38e-17 1.51e-11 1.01 8.70e-17 9.62e-11 1.05 8.91e-17 3.69e-10 0.48
1000 9.17e-17 1.63e-11 6.77 8.96e-17 5.38e-12 6.42 8.11e-17 6.93e-11 6.71 8.58e-17 2.82e-10 2.63
1500 8.66e-17 1.75e-11 18.49 8.76e-17 7.15e-12 18.40 8.94e-17 3.95e-11 18.49 8.76e-17 7.39e-11 7.69
2000 8.91e-17 1.52e-11 34.22 8.68e-17 6.17e-12 32.40 9.38e-17 4.12e-11 33.71 8.62e-17 2.07e-10 17.75
2500 9.45e-17 2.82e-11 63.48 9.18e-17 3.01e-12 60.78 7.82e-17 5.16e-11 62.92 8.66e-17 3.56e-10 33.42
3000 8.06e-17 1.70e-11 113.18 8.08e-17 8.29e-12 100.24 9.50e-17 3.59e-11 112.44 9.35e-17 6.29e-10 52.83
3500 8.56e-17 6.96e-12 169.34 1.01e-16 3.62e-12 162.04 1.02e-16 2.63e-11 162.79 7.56e-17 8.45e-11 81.62
4000 8.21e-17 1.34e-11 245.96 9.40e-17 3.91e-12 244.70 8.69e-17 3.41e-11 245.37 9.57e-17 1.38e-10 126.02
4500 7.84e-17 1.70e-11 351.13 9.31e-17 4.52e-12 313.81 9.19e-17 4.93e-11 328.67 8.42e-17 2.02e-10 175.69
5000 1.11e-16 3.03e-11 476.20 8.51e-17 4.20e-12 437.35 1.03e-16 5.77e-11 444.11 6.94e-17 2.67e-10 232.96
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