Analysis of dependence of grey wolf optimizer to shift-transformations and its shift-invariant improved methods adaptively controlling the search areas

Figures & data

Figure 1. Relations of ${\mathit{d}}^{(p,i)}$, ${\mathit{x}}^{\left(p\right)}$ and ${\mathit{x}}^{\left(i\right)}$.

Table 1. Benchmark problems.

Func.	Objective function and constraint	Optimal solution
$O1$: $2n$-minima	$\begin{array}{ll}min& \sum _{i=1}^n(x_i^4-16x_i^2+5x_i)-78.319n\\ \text{s.t.}& -5\le x_i\le 5,i\in N\end{array}$	$f\left({\mathit{x}}^{\ast }\right)=0$, ${\mathit{x}}^{\ast }=(-2.9,\dots ,-2.9{)}^{\mathrm{\top }}$
$O2$: Schwefel	$\begin{array}{ll}min& \sum _{i=1}^n-x_i\sin \left(\sqrt{\left|x_i\right|}\right)-418.98288727\times n\\ \text{s.t.}& -512\le x_i<512,i\in N\end{array}$	$f\left({\mathit{x}}^{\ast }\right)=0$, $\begin{array}{l}{\mathit{x}}^{\ast }=(420.968750,\dots ,\\ 420.968750{)}^{\mathrm{\top }}\end{array}$

Figure 2. Means of $\Vert \mathrm{\Delta }{\mathit{x}}^{(\alpha ,i)}\left(t\right)\Vert$, $\Vert {\mathit{d}}^{(\alpha ,i)}\left(t\right)\Vert$, and $f\left({\mathit{x}}^{\left(\alpha \right)}\right(t\left)\right)$ for four shift vectors.

Table 2. Comparison of results obtained by GWO for 20 benchmark problems (P1) and (P2) with three kinds of shift vectors ($\rho =0.1,0.5,0.9$).

	$\rho =0.0$		$\rho =0.1$		$\rho =0.5$		$\rho =0.9$
	Mean	SD	Mean	SD	Mean	SD	Mean	SD
O1	9.45E+02	1.55E+02	8.97E+02	1.56E+02	8.08E+02	1.07E+02	7.35E+02	1.25E+02
O2	1.47E+04	1.26E+03	1.44E+04	1.64E+03	1.41E+04	1.79E+03	1.22E+04	2.56E+03
C1	0.00E+00	0.00E+00	1.40E+08	5.83E+07	3.03E+09	1.19E+09	7.16E+09	2.48E+09
C2	0.00E+00	0.00E+00	1.39E+40	6.90E+40	1.52E+52	7.30E+52	1.13E+61	3.85E+61
C3	0.00E+00	0.00E+00	2.51E+02	9.33E+01	5.97E+03	1.72E+03	1.47E+04	5.48E+03
C4	4.55E+01	7.82E−01	5.46E+01	2.76E+00	2.63E+02	8.43E+01	5.81E+02	2.56E+02
C5	0.00E+00	0.00E+00	5.62E+01	1.27E+01	1.30E+02	2.12E+01	1.99E+02	2.50E+01
C6	1.33E+01	4.10E+00	1.32E+01	4.49E+00	1.81E+01	1.90E+00	2.00E+01	1.06E+00
C7	2.15E+00	2.66E−01	5.78E+01	1.74E+01	8.24E+02	1.90E+02	3.27E+03	2.20E+03
C8	1.82E−11	0.00E+00	1.10E+03	3.90E+02	5.01E+03	1.51E+03	5.02E+03	1.12E+03
C9	0.00E+00	0.00E+00	1.93E+06	1.19E+06	3.64E+07	1.76E+07	9.40E+07	5.14E+07
C10	0.00E+00	0.00E+00	6.77E+02	1.66E+02	1.33E+04	2.76E+03	3.32E+04	1.02E+04
C11	2.05E+01	5.54E−02	2.05E+01	5.44E−02	2.05E+01	5.03E−02	2.05E+01	6.35E−02
C12	0.00E+00	0.00E+00	6.28E+00	1.25E+00	1.68E+01	3.15E+00	2.37E+01	3.60E+00
C13	0.00E+00	0.00E+00	2.31E+00	5.82E−01	2.80E+01	1.11E+01	6.76E+01	3.11E+01
C14	2.18E−02	5.15E−03	2.05E−02	4.53E−03	2.38E−02	4.60E−03	2.45E−02	6.12E−03
C15	5.80E−01	6.45E−02	5.60E−01	7.21E−02	5.41E−01	7.21E−02	7.02E−01	1.96E−01
C16	4.11E−01	6.45E−02	3.76E−01	7.86E−02	1.67E+00	3.87E+00	1.66E+01	1.09E+01
C17	1.85E+01	2.51E+00	1.47E+01	5.19E+00	2.57E+02	2.13E+02	8.56E+03	1.41E+04
C18	5.09E−01	1.25E+00	9.24E−01	1.15E+00	2.43E+00	3.42E+00	2.77E+00	4.83E+00

Figure 3. Relation between ${\hat{d}}^{(p,i)}\left(t\right)$ and $v^{\left(p\right)}\left(t\right)$ when $c_m=3$ and $u^{(p,i)}=1.5$.

Table 3. Comparison of results obtained by GWO-SRs with seven different reference points and GWO-R with ${\mathit{\mu }}^{(1,2)}$ for 20 benchmark problems (P1).

Prob.	GWO-SR							GWO-R
No.	${\mathit{\mu }}^{\left(1\right)}$	${\mathit{\mu }}^{\left(2\right)}$	${\mathit{\mu }}^{\left(3\right)}$	${\mathit{\mu }}^{(1,2)}$	${\mathit{\mu }}^{(2,1)}$	${\mathit{\mu }}^{(1,3)}$	${\mathit{\mu }}^{(3,1)}$	${\mathit{\mu }}^{(1,2)}$
O1	3.99E+02	4.25E+02	4.33E+02	3.86E+02	4.14E+02	3.94E+02	4.52E+02	4.24E+02
O2	6.54E+03	6.82E+03	6.80E+03	6.49E+03	6.56E+03	6.47E+03	6.31E+03	6.08E+03
C1	1.79E−11	2.78E−09	1.60E+09	1.52E−14	4.19E−14	4.56E−12	1.24E−06	7.41E−09
C2	2.35E+50	1.45E+62	2.04E+72	1.46E+43	2.53E+60	1.36E+37	1.41E+72	1.94E+46
C3	4.34E−08	9.39E+02	9.27E+03	4.00E−06	1.18E+02	3.56E−06	8.95E+02	1.61E−03
C4	5.19E+01	7.21E+01	4.05E+02	5.40E+01	7.48E+01	5.31E+01	8.01E+01	5.67E+01
C5	7.84E+01	1.94E+02	2.16E+02	7.51E+01	1.83E+02	8.02E+01	2.07E+02	1.19E+02
C6	1.72E+01	1.76E+01	1.85E+01	1.77E+01	1.73E+01	1.76E+01	1.88E+01	1.80E+01
C9	1.53E+03	8.77E+03	9.70E+03	1.98E+03	8.19E+03	1.96E+03	8.00E+03	3.00E+03
C10	2.82E+03	5.40E+03	6.13E+03	2.77E+03	4.97E+03	2.98E+03	5.80E+03	3.53E+03
C11	4.50E−15	4.70E−09	3.99E+07	9.60E−18	4.04E−13	2.31E−15	2.04E−04	1.16E−10
C12	7.26E−16	8.01E−16	1.38E+04	5.06E−20	1.44E−19	1.57E−18	1.48E−08	2.87E−14
C13	2.06E+01	2.01E+01	2.00E+01	2.06E+01	2.05E+01	2.06E+01	2.02E+01	2.09E+01
C14	7.84E−01	3.91E+00	7.39E+00	8.72E−01	3.39E+00	1.06E+00	6.56E+00	4.72E+00
C15	2.81E−02	3.69E−02	2.00E+01	1.95E−02	5.58E−02	1.17E−02	1.13E−01	3.29E−02
C16	4.70E−02	2.27E−17	0.00E+00	4.80E−02	1.02E−03	4.69E−02	1.08E−17	9.05E−02
C17	4.48E−01	5.40E−01	6.01E−01	4.89E−01	5.30E−01	4.51E−01	5.91E−01	4.93E−01
C18	6.12E−01	6.04E−01	5.10E+00	6.33E−01	6.33E−01	5.54E−01	6.25E−01	5.18E−01
C19	1.41E+01	6.36E+01	1.12E+04	1.92E+01	3.03E+01	2.35E+01	4.85E+01	1.84E+01
C20	4.67E−02	1.14E−01	3.06E−01	4.48E−02	5.24E−02	6.21E−02	1.24E−01	4.85E−02

Table 4. Comparison of PSO, FA, GWO-SR and GWO-AS for 50-dimensional problems ($f_{max}=1200000$).

	PSO		FA		GWO-SR		GWO-AS
	Mean	SD	Mean	SD	Mean	SD	Mean	SD
O1	5.50E+02	9.39E+01	5.81E+02	8.96E+01	3.92E+02	9.09E+01	1.35E+02	6.89E+01
O2	7.05E+03	1.06E+03	1.58E+04	1.86E+03	6.67E+03	6.72E+02	3.41E+03	4.59E+02
C1	2.45E−179	0.00E+00	7.61E+06	4.61E+05	9.73E−62	5.25E−61	1.64E−22	3.88E−22
C2	2.00E+40	1.41E+41	0.00E+00	0.00E+00	1.79E+22	1.25E+23	8.80E+10	6.22E+11
C3	5.93E−165	0.00E+00	8.54E+00	5.26E−01	2.25E−34	1.23E−33	6.54E−14	2.68E−13
C4	1.42E+01	5.93E+00	4.31E+01	5.46E−01	4.52E+01	2.45E+01	4.17E+01	1.51E+00
C5	1.22E+02	3.32E+01	2.35E+01	4.64E+00	6.20E+01	1.36E+01	1.20E+01	3.27E+00
C6	1.48E+01	1.35E+00	9.85E+00	7.86E−01	1.58E+01	1.75E+00	2.56E+00	5.96E−01
C9	8.01E+02	6.37E+02	4.18E+02	8.88E+01	1.53E+03	1.10E+03	2.64E+00	3.12E+00
C10	3.80E+03	7.13E+02	1.32E+03	5.00E+02	2.24E+03	7.12E+02	3.12E+02	2.11E+02
C11	6.70E+03	3.24E+04	2.67E+05	5.50E+04	1.33E−65	5.51E−65	1.38E−25	7.61E−25
C12	6.15E−185	0.00E+00	1.84E+03	9.70E+02	4.46E−67	2.16E−66	4.73E−28	1.78E−27
C13	2.00E+01	2.25E−03	3.07E+00	5.32E−02	2.06E+01	5.52E−02	2.02E+01	5.60E−02
C14	1.14E+01	3.63E+00	1.50E+01	2.64E+00	2.56E−01	5.56E−01	1.48E−02	2.44E−02
C15	2.47E−02	3.31E−02	1.07E+00	4.41E−03	6.71E−03	1.59E−02	4.53E−03	6.63E−03
C16	6.68E−05	2.26E−04	2.23E−01	2.79E−02	3.68E−02	1.46E−02	5.15E−03	1.70E−03
C17	5.81E−01	9.65E−02	7.30E−01	1.31E−01	4.07E−01	8.12E−02	4.61E−01	7.61E−02
C18	5.76E−01	2.58E−01	5.05E−01	9.97E−02	6.25E−01	2.91E−01	4.99E−01	2.78E−01
C19	1.23E+01	3.38E+00	2.27E+01	1.47E+00	1.01E+01	7.83E+00	4.68E+00	9.54E−01
C20	2.81E−02	4.09E−02	2.60E−01	9.20E−02	2.51E−02	4.41E−02	4.04E−03	5.05E−03

Table 5. Comparison of PSO, FA, GWO-SR and GWO-AS for 200-dimensional problems ( $f_{max}=4800000$ ).

	PSO		FA		GWO-SR		GWO-AS
	Mean	SD	Mean	SD	Mean	SD	Mean	SD
O1	2.46E+03	2.22E+02	2.50E+03	1.36E+02	2.25E+03	2.17E+02	1.20E+03	1.73E+02
O2	3.29E+04	2.67E+03	6.57E+04	6.27E+03	2.67E+04	1.41E+03	1.06E+04	7.98E+02
C1	3.00E+03	4.66E+03	6.15E+07	1.42E+06	9.46E−18	5.18E−17	1.02E−05	7.35E−06
C3	1.68E−23	9.22E−23	7.41E+01	4.76E+00	1.66E−03	5.92E−03	2.42E−02	5.47E−02
C4	1.86E+02	3.55E+01	1.90E+02	5.45E−01	3.56E+02	6.83E+01	2.44E+02	5.04E+01
C5	7.52E+02	7.90E+01	1.17E+02	2.47E+01	7.23E+02	8.25E+01	5.38E+01	7.69E+00
C6	7.50E+01	2.08E+00	5.57E+01	4.96E+00	8.44E+01	4.18E+00	1.45E+01	2.04E+00
C9	1.49E+04	2.27E+03	2.58E+03	5.09E+02	9.33E+04	4.85E+04	3.50E+01	1.96E+01
C10	2.19E+04	1.52E+03	7.30E+03	2.03E+03	2.09E+04	1.69E+03	3.16E+03	5.08E+02
C11	1.15E+06	2.34E+06	3.06E+06	2.16E+05	1.18E−18	6.48E−18	9.39E−09	1.14E−08
C12	1.33E+03	4.34E+03	1.05E+03	1.03E+04	1.12E−27	1.53E−27	1.11E−11	1.11E−11
C13	2.00E+01	3.12E−04	3.77E+00	3.56E−02	2.10E+01	3.30E−02	2.05E+01	3.58E−02
C14	1.43E+02	1.24E+01	7.37E+01	7.47E+00	3.97E+01	8.50E+00	1.64E+00	5.10E−01
C15	8.65E−01	1.22E+00	1.56E+00	1.15E−02	2.83E−01	3.95E−01	9.44E−02	1.48E−01
C16	4.38E−06	9.81E−06	2.52E−01	1.31E−01	3.59E−02	4.81E−03	1.64E−03	2.66E−04
C17	7.89E−01	7.33E−02	9.21E−01	3.88E−02	6.69E−01	4.63E−02	7.21E−01	5.58E−02
C18	5.84E−01	2.96E−01	5.29E−01	8.33E−02	5.17E−01	2.94E−01	6.74E−01	2.71E−01
C19	3.33E+02	1.01E+02	1.19E+02	6.85E+00	1.84E+03	6.56E+02	3.06E+01	4.59E+00
C20	7.93E−02	5.53E−02	7.21E−01	2.15E−01	9.46E−02	4.66E−02	1.57E−02	1.05E−02

Figure 4. Means of the best function values obtained by four methods, PSO, FA, GWO-SR and GWO-AS, until function evaluation count per agent for 50-dimensional six benchmark problems.

Figure 5. Means of $\left|W^{\left(k\right)}\right(t+1\left)\right|$, $k\in \{1,2,3\}$ for three candidates on 50 trials at each count when GWO-AS is applied to 50-dimensional four benchmark problems.