Sufficient Conditions for Global Convergence of Differential Evolution Algorithm

Table 4

The function error values achieved via CCoDE-umbest and CoDE on CEC2005's test functions.

FES	1.5𝐸+5				3.0𝐸+5
Alg.	CCoDE-umbest		CoDE		CCoDE-umbest		CoDE
	Mean	Std.	Mean	Std.	Mean	Std.	Mean	Std.
f1	0.00𝐸+00	0.00𝐸+00	0.00𝐸+00	0.00𝐸+00	0.00𝐸+00	0.00𝐸+00	0.00𝐸+00	0.00𝐸+00
f2	1.83𝐸−04	1.21𝐸−04	5.16𝐸−04	2.28𝐸−04	1.14𝐸−11	1.13𝐸−11	7.42𝐸−11	8.58𝐸−11
f3	1.01𝐸+04	9.71𝐸+03	2.78𝐸+05	1.07𝐸+05	5.33𝐸+03	9.45𝐸+03	5.05𝐸+04	1.34𝐸+04
f4	8.19𝐸−03	5.36𝐸−03	1.84𝐸−02	1.07𝐸−02	1.67𝐸−08	1.39𝐸−08	7.72𝐸−08	6.33𝐸−08
f5	2.98𝐸−03	1.37𝐸−03	8.78𝐸−02	3.06𝐸−02	9.56𝐸−10	5.66𝐸−10	1.11𝐸−06	7.15𝐸−07
f6	2.88𝐸−05	2.40𝐸−05	5.62𝐸−03	3.56𝐸−03	3.41𝐸−14	3.22𝐸−14	5.91𝐸−09	4.29𝐸−09
f7	2.12𝐸−01	1.46𝐸−01	2.69𝐸−01	5.34𝐸−02	2.12𝐸−01	1.46𝐸−01	1.69𝐸−01	3.94𝐸−02
f8	2.01𝐸+01	7.89𝐸−02	2.03𝐸+01	5.47𝐸−02	2.01𝐸+01	7.08𝐸−02	2.03𝐸+01	6.77𝐸−02
f9	0.00𝐸+00	0.00𝐸+00	0.00𝐸+00	0.00𝐸+00	0.00𝐸+00	0.00𝐸+00	0.00𝐸+00	0.00𝐸+00
f10	8.81𝐸+00	3.05𝐸+00	2.05𝐸+01	3.25𝐸+00	8.80𝐸+00	3.05𝐸+00	1.40𝐸+01	2.52𝐸+00
f11	1.59𝐸+00	9.51𝐸−01	6.17𝐸+00	6.60𝐸−01	1.16𝐸+00	1.01𝐸+00	5.28𝐸+00	8.16𝐸−01
f12	2.76𝐸+00	4.89𝐸+00	5.00𝐸+02	2.03𝐸+02	1.64𝐸+00	4.78𝐸+00	1.54𝐸+02	1.14𝐸+02
f13	3.40𝐸−01	1.63𝐸−01	7.11𝐸−01	8.70𝐸−02	3.12𝐸−01	1.48𝐸−01	5.01𝐸−01	6.40𝐸−02
f14	3.03𝐸+00	3.50𝐸−01	3.35𝐸+00	1.90𝐸−01	3.03𝐸+00	3.50𝐸−01	3.26𝐸+00	1.73𝐸−01
f15	0.00𝐸+00	0.00𝐸+00	3.24𝐸+00	1.10𝐸+01	0.00𝐸+00	0.00𝐸+00	3.24𝐸+00	1.10𝐸+01
f16	1.10𝐸+02	7.15𝐸+00	1.34𝐸+02	5.87𝐸+00	1.10𝐸+02	7.20𝐸+00	1.16𝐸+02	5.68𝐸+00
f17	1.10𝐸+02	9.83𝐸+00	1.55𝐸+02	1.02𝐸+01	1.09𝐸+02	9.71𝐸+00	1.38𝐸+02	7.95𝐸+00
f18	4.80𝐸+02	2.40𝐸+02	5.00𝐸+02	2.45𝐸+02	4.80𝐸+02	2.40𝐸+02	5.00𝐸+02	2.45𝐸+02
f19	5.80𝐸+02	2.48𝐸+02	4.20𝐸+02	2.14𝐸+02	5.80𝐸+02	2.48𝐸+02	4.20𝐸+02	2.14𝐸+02
f20	5.40𝐸+02	2.50𝐸+02	4.00𝐸+02	2.00𝐸+02	5.40𝐸+02	2.50𝐸+02	4.00𝐸+02	2.00𝐸+02
f21	5.16𝐸+02	7.84𝐸+01	5.00𝐸+02	3.41𝐸−13	5.16𝐸+02	7.84𝐸+01	5.00𝐸+02	3.41𝐸−13
f22	6.80𝐸+02	1.66𝐸+02	6.20𝐸+02	2.19𝐸+02	6.79𝐸+02	1.66𝐸+02	6.16𝐸+02	2.17𝐸+02
f23	5.59𝐸+02	1.46𝐸−12	5.59𝐸+02	1.32𝐸−12	5.59𝐸+02	1.46𝐸−12	5.59𝐸+02	3.41𝐸−13
f24	2.00𝐸+02	0.00𝐸+00	2.00𝐸+02	0.00𝐸+00	2.00𝐸+02	0.00𝐸+00	2.00𝐸+02	0.00𝐸+00
f25	2.00𝐸+02	0.00𝐸+00	2.00𝐸+02	0.00𝐸+00	2.00𝐸+02	0.00𝐸+00	2.00𝐸+02	0.00𝐸+00
Sign test on mean	Neg_Dif	Pos_Dif	Ties	P value	Neg_Dif	Pos_Dif	Ties	P value
	16	4	5	0.012	15	5	5	0.041

“FES” denotes the number of function evaluations. “Std.” denotes the standard deviation of 25 mean errors. The two bottom lines record the test statistics for sign test on mean errors. “Neg_Dif” and “Pos_Dif” denote the number of negative and positive differences, respectively. “P value” denotes the probability value supporting the null hypothesis. Here the both P value are less than the significance level at 0.05.