Enhanced Differential Evolution Algorithm with Local Search Based on Hadamard Matrix

Table 2

Experimental results of four classical operators of DE with HLS for all test functions and comparison without HLS.

Fun	DE/rand/1/bin	DE/rand/1/bin + HLS	DE/best/1/bin	DE/best/1/bin + HLS	DE/rand-to-best/1/bin	DE/rand-to-best/1/bin + HLS	DE/current-to-best/1/bin	DE/current-to-best/1/bin + HLS
	Mean error ± Std error	Mean error ± Std error	Mean error ± Std error	Mean error ± Std error	Mean error ± Std error	Mean err ± Std err	Mean err ± Std err	Mean err ± Std err
f1	3.47e − 18 ± 7.66e − 18−	1.12E − 112 ± 6.03E − 112	1.94e − 12 ± 1.06e − 11-	2.94E − 20 ± 1.61E − 19	3.41e − 106 ± 1.85e − 105+	1.70E − 49 ± 9.31E − 49	4.90e − 108 ± 1.89e − 107+	2.90E − 17 ± 1.59E − 16
f2	6.46e − 11 ± 9.73e − 11−	2.30E − 58 ± 1.22E − 57	1.90e − 04 ± 1.04e − 03≈	7.53E − 04 ± 4.09E − 03	1.83e − 59 ± 7.68e − 59+	1.36E − 06 ± 7.47E − 06	3.55e − 61 ± 1.27e − 60+	3.96E − 12 ± 2.17E − 11
f3	3.46e − 02 ± 3.26e − 02+	1.65E + 00 ± 2.73E + 00	1.55e − 21 ± 4.76e − 21+	2.10E − 15 ± 7.15E − 15	2.86e − 24 ± 7.23e − 24+	3.93E − 17 ± 6.83E − 17	6.53e − 25 ± 1.50e − 24+	2.39E + 00 ± 1.31E + 01
f4	1.53e + 01 ± 7.28e + 00−	1.73E − 165 ± 0.00E + 00	3.60e + 01 ± 7.31e + 00−	2.70E − 03 ± 1.45E − 02	2.23e + 01 ± 5.68e + 00−	3.72E − 175 ± 0.00E + 00	2.63e + 01 ± 7.30e + 00−	5.02E − 05 ± 2.12E − 04
f5	8.83e + 00 ± 3.21e + 00≈	1.19E + 01 ± 1.38E + 01	9.30e − 01 ± 1.71e + 00+	2.27E + 01 ± 1.16E + 01	1.20e + 00 ± 1.86e + 00+	2.55E + 01 ± 8.66E + 00	1.46e + 00 ± 1.95e + 00+	2.27E + 01 ± 1.16E + 01
f6	2.67e − 01 ± 7.85e − 01−	0.00E + 00 ± 0.00E + 00	2.27e + 02 ± 5.80e + 02−	0.00E + 00 ± 0.00E + 00	3.03e + 01 ± 7.46e + 01−	0.00E + 00 ± 0.00E + 00	1.85e + 01 ± 2.45e + 01−	0.00E + 00 ± 0.00E + 00
f7	1.63e − 02 ± 5.98e − 03−	3.54E − 03 ± 4.00E − 03	1.11e − 02 ± 7.48e − 03-	2.30E − 03 ± 4.55E − 03	8.50e − 03 ± 4.20e − 03−	1.70E − 03 ± 3.10E − 03	8.40e − 03 ± 4.20e − 03−	1.30E − 03 ± 1.36E − 03
f8	1.32e + 04 ± 2.47e + 02−	1.26E + 04 ± 0.00E + 00	1.55e + 04 ± 6.80e + 02−	1.38E + 04 ± 1.77E + 03	1.49e + 04 ± 5.66e + 02−	1.32E + 04 ± 1.29E + 03	1.54e + 04 ± 6.62e + 02−	1.30E + 04 ± 1.13E + 03
f9	2.41e + 01 ± 7.72e + 00−	0.00E + 00 ± 0.00E + 00	6.60e + 01 ± 2.03e + 01−	1.49E + 01 ± 2.40E + 01	3.66e + 01 ± 8.98e + 00−	2.78E + 00 ± 8.54E + 00	4.09e + 01 ± 1.12e + 01−	6.41E + 00 ± 1.29E + 01
f10	3.68e − 10 ± 3.02e − 10−	4.74E − 16 ± 1.23E − 15	4.78e + 00 ± 2.44e + 00−	3.46E − 01 ± 1.51E + 00	2.34e + 00 ± 1.35e + 00−	3.04E − 14 ± 1.59E − 13	2.78e + 00 ± 2.04e + 00−	2.01E − 15 ± 2.02E − 15
f11	3.37e − 03 ± 8.55e − 03−	0.00E + 00 ± 0.00E + 00	5.09e − 02 ± 7.59e − 02−	1.58E − 02 ± 6.72E − 02	2.55e − 02 ± 3.08e − 02−	3.11E − 03 ± 1.08E − 02	1.55e − 02 ± 1.95e − 02-	5.75E − 04 ± 2.21E − 03
f12	4.53e − 02 ± 2.29e − 01−	1.57E − 32 ± 1.91E − 34	1.01e + 00 ± 2.15e + 00−	3.46E − 03 ± 1.89E − 02	5.00e − 01 ± 8.89e − 01−	1.73E − 02 ± 7.74E − 02	7.25e − 01 ± 1.00e + 00−	1.68E − 32 ± 2.47E − 33
f13	7.32e − 04 ± 2.79e − 03−	1.35E − 32 ± 5.57E − 48	7.44e − 01 ± 1.70e + 00−	3.66E − 04 ± 2.01E − 03	5.65e − 01 ± 1.49e + 00−	4.01E − 32 ± 8.12E − 32	3.61e − 01 ± 9.67e − 01−	3.66E − 04 ± 2.01E − 03
f14	1.65e − 18 ± 4.20e − 18−	0.00E + 00 ± 0.00E + 00	8.77e − 22 ± 4.80e − 21+	1.62E + 01 ± 8.89E + 01	5.07e − 30 ± 9.73e − 30+	4.80E − 05 ± 2.63E − 04	3.57e − 30 ± 2.69e − 30+	4.16E − 17 ± 2.27E − 16
f15	3.73e − 02 ± 3.54e − 02+	1.16E + 00 ± 1.75E + 00	5.48e − 22 ± 1.45e − 21+	1.05E − 15 ± 1.88E − 15	5.30e − 24 ± 2.06e − 23+	1.12E − 06 ± 6.14E − 06	8.67e − 25 ± 2.10e − 24+	7.27E − 17 ± 1.80E − 16
f16	3.73e − 02 ± 4.47e − 02+	2.76E − 01 ± 5.10E − 01	9.76e − 03 ± 4.36e − 02−	1.42E − 05 ± 4.99E − 05	3.02e − 07 ± 1.17e − 06+	3.75E + 00 ± 2.05E + 01	2.75e − 10 ± 1.34e − 09+	8.68E − 07 ± 4.75E − 06
f17	1.81e − 03 ± 4.92e − 03−	0.00E + 00 ± 0.00E + 00	4.13e − 02 ± 4.73e − 02−	7.75E − 03 ± 4.24E − 02	1.65e − 02 ± 1.65e − 02−	2.47E − 04 ± 1.35E − 03	3.13e − 02 ± 3.18e − 02−	0.00E + 00 ± 0.00E + 00
f18	3.13e − 10 ± 2.45e − 10−	3.55E − 16 ± 1.08E − 15	3.25e + 00 ± 1.95e + 00−	1.78E − 03 ± 9.72E − 03	2.31e + 00 ± 1.19e + 00−	1.18E − 15 ± 1.94E − 15	2.62e + 00 ± 9.67e − 01−	1.66E − 15 ± 1.80E − 15
f19	6.91e − 03 ± 3.79e − 02−	1.57E − 32 ± 1.77E − 34	9.70e − 01 ± 1.87e + 00−	1.71E − 32 ± 3.20E − 33	4.53e − 01 ± 6.95e − 01−	1.66E − 32 ± 1.91E − 33	6.54e − 01 ± 9.23e − 01−	3.46E − 03 ± 1.89E − 02
f20	1.19e − 17 ± 4.41e − 17−	1.35E − 32 ± 5.57E − 48	1.73e + 04 ± 9.47e + 04−	1.96E − 31 ± 4.78E − 31	2.38e − 01 ± 5.94e − 01−	6.13E − 32 ± 2.62E − 31	4.67e − 01 ± 1.63e + 00−	6.79E − 32 ± 2.55E − 31
w/b/s	16/3/1		15/4/1		13/7/0		13/7/0