An Improved Atomic Search Algorithm

doi:10.16182/j.issn1004731x.joss.20-0824

Abstract

Abstract:

The atom search algorithm (ASO) is a new optimization algorithm proposed by imitating the movement of atoms in the natural world. An improved atomic search algorithm (IASO) is proposed to address the problems of prematureness and slow convergence of ASO in solving complex functions. IASO adds the binding force generated by the historical optimal solution of individual atoms to correct the acceleration of ASO and enhance the global search capability. The two multiplier coefficients are adaptively updated to coordinate the algorithm's global search and local development capabilities. The Gaussian mutation strategy is used to re-update the atomic position and improve the ability to jump out of precocity. Carrying out simulation experiments on 14 benchmark functions and comparing other algorithms, IASO shows superior performance in terms of convergence speed and convergence accuracy.

Key words: atomic optimization algorithm, function optimization, self-adaptation, Gaussian mutation, convergence accuracy, test function

CLC Number:

TP301.6

Jianfeng Li, Di Lu, Hexiang Li. An Improved Atomic Search Algorithm[J]. Journal of System Simulation, 2022, 34(3): 490-502.

Figures/Tables 9

Fig. 1

Fig. 2

Table 1

Table 2

Benchmark test functions

Benchmark function	Dim	Feature	Range	f_min
$f 1 x = ∑ i = 1 n x i 2$	30	[-100,100]	[-100,100]	0
$f 2 x = ∑ i = 1 n x i + ∏ i = 1 n x i$	30	[-10,10]	[-10,10]	0
$f 3 x = ∑ i = 1 n ∑ j = 1 i x j 2$	30	[-100,100]	[-100,100]	0
$f 4 x = m a x i x i, 1 ≤ i ≤ D$	30	[-100,100]	[-100,100]	0
$f 5 x = ∑ i = 1 n x i + 0.5 2$	30	[-100,100]	[-100,100]	0
$f 6 x = ∑ i = 1 n x i 4 + r a n d 0,1$	30	[-1.28,1.28]	[-1.28,1.28]	0
$f 7 x = ∑ i = 1 n - x i s i n x i$	30	[-500,500]	[-500,500]	-418.982 9×5
$f 8 x = ∑ i = 1 n x i 2 - 10 c o s 2 π x i + 10$	30	[-5.12,5.12]	[-5.12,5.12]	0
$f 9 x = - 20 e x p - 0.2 1 n ∑ i - 1 n x i 2 - e x p 1 n ∑ i = 1 n c o s 2 π x i + 20 + e$	30	[-32,32]	[-32,32]	0
$f 10 x = 1 4 ? 000 ∑ i = 1 n x i 2 - ∏ i = 1 n c o s x i i + 1$	30	[-600,600]	[-600,600]	0
$f 11 x = 0.1 s i n 2 3 π x 1 + ∑ i - 1 n x i - 1 2 1 + s i n 2 3 π x 1 + 1 + x n - 1 2 1 + s i n 2 2 π x n + ∑ i = 1 n u x i, 5,100,4$	30	[-50,50]	[-50,50]	0
$f 12 x = 1500 + ∑ j = 1 25 1 j + ∑ i = 1 2 x i - a i j 6 - 1$	2	[-65,65]	[-65,65]	1
$f 13 x = ∑ i = 1 11 a i - x 1 b i 2 + b i x 2 b i 2 + b i x 3 + x 4 2$	4	[-5,5]	[-5,5]	0.000 30
$f 14 = - ∑ i = 1 7 X - a i X - a i T + C i - 1$	4	[0.10]	[0.10]	-10.402 8

Table 2

Fig. 3

Fig. 4

Table 3

Experimental results of test function

Function	Result	Algorithm
Function	Result	GWO	MFO	WOA	ASO	IASO
$f 1$	Best	0	0.000 1E-11	0	0.003 6E-21	0
	Worst	0.786 4E-58	0.144 1E-11	0.907 2E-80	0.100 0E-21	0.166 5E-110
	Mean	0.045 1E-58	0.028 5E-11	0.030 2E-80	0.034 6E-21	0.005 6E-110
	Std	0.146 0E-58	0.041 2E-11	0.165 6E-80	0.027 8E-21	0.030 4E-110
$f 2$	Best	0.000 3E-31	0.003 8E-7	0	0.008 0E-9	0
	Worst	0.365 6E-31	0.206 8E-7	0.171 2E-52	0.168 4E-9	0.777 5E-80
	Mean	0.054 3E-31	0.055 3E-7	0.016 9E-52	0.043 6E-9	0.025 9E-80
	Std	0.083 3E-31	0.053 7E-7	0.042 7E-52	0.032 0E-9	0.141 9E-80
$f 3$	Best	0	0	1.484 2	1.176 7E-17	0
	Worst	0.101 7E-27	0.005 1	6.981 9	0.007 8E-17	0.285 1E-142
	Mean	0.008 2E-27	0.008 3	4.309 4	0.000 4E-17	0.018 1E-142
	Std	0.023 4E-27	0.019 1	1.171 1	0.001 5E-17	0.068 9E-142
$f 4$	Best	0	0.001 6	0.002 5	0.070 3E-11	0
	Worst	0.196 2E-16	1.551 9	16.573 5	0.992 3E-11	0.654 1E-72
	Mean	0.021 7E-16	0.243 9	1.383 9	0.346 1E-11	0.021 8E-72
	Std	0.041 4E-16	0.296 7	3.447 6	0.249 7E-11	0.119 4E-72
$f 5$	Best	0	0	0.000 1	0	0
	Worst	0.252 2E-5	0.152 5E-12	0.010 3	0.116 7E-22	0.654 5E-22
	Mean	0.016 8E-5	0.236 0E-12	0.001 1	0.033 8E-22	0.205 1E-22
	Std	0.063 8E-5	0.033 3E-12	0.001 8E-2	0.029 9E-22	0.172 3E-22
$f 6$	Best	0.000 1E-3	0.003 3	0.013 4E-2	0.002 7	0.263 3E-4
	Worst	0.001 5E-3	0.024 4	0.003 0E-2	0.043 0	3.147 1E-4
	Mean	0.000 6E-3	0.009 3	0.002 9E-2	0.013 5	1.204 0E-4
	Std	0.000 4E-3	0.005 9	0.389 3	0.009 2	0.745 0E-4
$f 7$	Best	-3.182 7E+3	-3.833 0E+3	-4.189 6E+3	-3.538 4E+3	-3.071 4E+3
	Worst	-2.045 6E+3	-2.521 1E+3	-2.199 6E+3	-2.373 7E+3	-2.886 5E+3
	Mean	-2.702 4E+3	-3.213 9E+3	-3.467 7E+3	-2.783 6E+3	-3.587 0E+3
	Std	0.293 2E+3	0.339 6E+3	0.637 6E+3	0.283 7E+3	0.278 8E+3
$f 8$	Best	0	4.974 8	0	1.989 9	0
	Worst	6.486 2E-13	78.743 5	4.175 1E-13	16.914 3	0
	Mean	1.103 1E-13	25.316 9	3.180 2E-13	8.656 1	0
	Std	2.116 5E-13	15.587 8	1.043 9E-13	3.223 0	0
$f 9$	Best	0.444 1E-14	0	0.008 9E-13	0.070 0E-10	0.888 2E-15
	Worst	0.799 4E-14	19.950 2E-6	0.151 0E-13	0.433 8E-10	0.888 2E-15
	Mean	0.680 9E-14	0.719 9E-6	0.050 3E-13	0.197 5E-10	0.888 2E-15
	Std	0.170 3E-14	3.644 4E-6	0.033 7E-13	0.087 5E-10	0
$f 10$	Best	0	0.017 2	0	0	0
	Worst	0.094 5E-14	0.666 4	0.330 7E-14	0.027 0E-13	0
	Mean	0.024 6E-14	0.155 5	0.048 3E-14	0.004 4E-13	0
	Std	0.021 4E-14	0.133 2	0.087 5E-14	0.007 8E-13	0
$f 11$	Best	0.000 0	0.000 0	0.000 8	0.004 4	0
	Worst	0.201 5	0.011 0	0.211 3	0.302 1	0.804 3E-22
	Mean	0.022 9	0.003 3	0.042 2	0.055 0	0.211 1E-22
	Std	0.049 9	0.005 1	0.049 5	0.061 0	0.190 8E-22
$f 12$	Best	0.998 0	0.998 0	0.998 0	0.998 0	0.998 0
	Worst	12.670 5	7.874 0	10.763 2	0.001 4	1.992 0
	Mean	3.676 0	2.249 2	3.025 8	3.968 9	1.064 5
	Std	3.875 0	1.918 4	3.354 9	1.398 4	0.252 2
$f 13$	Best	0.000 3	0.000 6	0.000 3	0.716 2	0.000 3
	Worst	0.056 6	0.020 4	0.001 5	0.001 4	0.000 7
	Mean	0.005 7	0.002 1	0.000 7	0.001 0	0.000 5
	Std	0.012 2	0.003 9	0.000 3	0.000 2	0.000 1
$f 14$	Best	-10.402 7	-10.402 9	-10.401 0	-10.402 9	-10.402 9
	Worst	-5.127 3	-1.837 6	-2.765 3	-2.751 9	-5.128 8
	Mean	-10.225 8	-6.457 5	-8.493 2	-6.147 9	-10.227 1
	Std	0.962 9	3.095 1	2.653 7	1.396 9	0.962 9

Table 3

Fig. 5

Table 4

Wilcoxon rank sum test results

function	Significance level P
function	GWO	MFO	WOA	ASO
$f 1$	0	0	0	0
$f 2$	0	0	0	0
$f 3$	0	0	0	0
$f 4$	0	0	0	0
$f 5$	0	0	0	0.158 0
$f 6$	0	0	0	0
$f 7$	0	0.021 6	0.333 7	0
$f 8$	0	0	0	0
$f 9$	0.000 2	0.001 2	0.000 3	0.001 2
$f 10$	0	0	0.000 1	0
$f 11$	0	0	0	0
$f 12$	0	0.075	0	0.001 9
$f 13$	0.004 9	0	0.001 2	0

Table 4

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