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

Fakhrodin Hashemi; Saeed Ketabchi

doi:10.3934/naco.2019029

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2020, Volume 10, Issue 1: 13-21. Doi: 10.3934/naco.2019029

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

Fakhrodin Hashemi and
Saeed Ketabchi^,

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

^* Corresponding author: Saeed Ketabchi
^* Corresponding author: Saeed Ketabchi

Received: June 2018

Revised: April 2019

Published: March 2020

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Abstract

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.

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Mathematics Subject Classification: Primary: 65D10; Secondary: 34M03, 90C25, 90C26.

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Figure 1. Performance profile of computing time of Algorithm 2 for the example 1

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Figure 2. Performance profile of computing time of Algorithm 2 for the example 2

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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

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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

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