Antlion optimizer algorithm based on chaos search and its application

doi:10.21629/JSEE.2019.02.14

Journal of Systems Engineering and Electronics ›› 2019, Vol. 30 ›› Issue (2): 352-365.doi: 10.21629/JSEE.2019.02.14

• Systems Engineering • Previous Articles Next Articles

Antlion optimizer algorithm based on chaos search and its application

Zhenxing ZHANG¹(), Rennong YANG¹(), Huanyu LI^1,*(), Yuhuan FANG²(), Zhenyu HUANG¹(), Ying ZHANG¹()

¹ Air Traffic Control and Navigation College, Air Force Engineering University, Xi'an 710038, China
² Unit 95939 of PLA, Cangzhou 061736, China

Received:2017-12-04 Online:2019-04-01 Published:2019-04-28
Contact: Huanyu LI E-mail:2207621676@qq.com;786918169@qq.com;lihuanyu1984@163.com;846655874@qq.com;65486346@qq.com;zhangying198807@126.com
About author:ZHANG Zhenxing was born in 1993. He received his M.S. degree from Airforce Engineering University. He is a Ph.D. candidate in Airforce Engineering University. His research interests are artificial intelligence and deep learning algorithm. E-mail:2207621676@qq.com|YANG Rennong was born in 1969. He received his M.S. degree from Airforce Engineering University. He is a professor in Airforce Engineering University. His research interests are artificial intelligence and deep learning algorithm. E-mail:786918169@qq.com|LI Huanyu was born in 1984. He received his Ph.D. degree from Airforce Engineering University. He is a lecturer in Airforce Engineering University. His research interests are artificial intelligence and deep learning algorithm. E-mail:lihuanyu1984@163.com|FANG Yuhuan was born in 1994. He received her M.S. degree from Airforce Engineering University. She is an assistant engineer in Unit 95939 of PLA. Her research interests are artificial intelligence and deep learning algorithm. E-mail:846655874@qq.com|HUANG Zhenyu was born in 1989. He received his M.S. degree from Airforce Engineering University. He is a lecturer in Airforce Engineering University. His research interests are artificial intelligence and deep learning algorithm. E-mail:65486346@qq.com|ZHANG Ying was born in 1988. He received her Ph.D. degree from Airforce Engineering University. She is a lecturer in Airforce Engineering University. Her research interests are artificial intelligence and deep learning algorithm. E-mail:zhangying198807@126.com
Supported by:
the National Natural Science Foundation of China(61503409);the Aeronautical Science Foundation of China(20155196022);the Natural Science Foundation of Shanxi Province(2016JQ6050);This work was supported by the National Natural Science Foundation of China (61503409), the Aeronautical Science Foundation of China (20155196022) and the Natural Science Foundation of Shanxi Province (2016JQ6050)

Abstract

Abstract:

Aiming at the problems of premature convergence and easily falling into local optimums of the antlion optimization algorithm, a chaos antlion optimization algorithm based on the chaos search is proposed. Firstly, in the algorithm, the population is initialized by using the tent chaotic mapping, and the self-adaptive dynamic adjustment of chaotic search scopes is proposed in order to improve the overall fitness and the optimization efficiency of the population. Then, the tournament strategy is used to select antlions. Finally, the logistic chaos operator is used to optimize the random walk of ants, which forms a global and local parallel search mode with the antlionos foraging behavior. The performance algorithm is tested through 13 complex high-dimensional benchmark functions and three dimensional path planning problems. The experimental results of six complex high-dimensional benchmark functions show that the presented algorithm has a better convergence speed and precision than the standard antlion algorithm and other optimization algorithms, and is suitable for the optimization of complex high dimensional functions. The trajectory planning experimental results show that compared with the antlion optimizer (ALO) algorithm, grey wolf optimizer (GWO), particle swarm optimization (PSO) and artificial bee colony (ABC) algorithm, it has advantages in speed and accuracy to obtain a specific path, and it is of great value in actual problems.

Key words: antlion optimizer (ALO) algorithm, chaos theory, tent mapping, tournament selection strategy, logistic mapping, path planning

Zhenxing ZHANG, Rennong YANG, Huanyu LI, Yuhuan FANG, Zhenyu HUANG, Ying ZHANG. Antlion optimizer algorithm based on chaos search and its application[J]. Journal of Systems Engineering and Electronics, 2019, 30(2): 352-365.

Figures/Tables 19

Table 1

Benchmark functions"

Function	Expression and range	Global minimum	Characteristic
$f_1 $ Sphere	$f({{{\mathit{\boldsymbol{X}}}}}) = { }\sum\limits_{i = 1}^D x_i^2, \; x_i \in [-100, 100]$	$\begin{array}{l} x_i = 0, \\ f({{{\mathit{\boldsymbol{X}}}}}) = 0 \\ \end{array}$	Unimodal
$f_2 $ Rosenbrock	$f({{{\mathit{\boldsymbol{X}}}}}) = { }\sum\limits_{i = 1}^D \|x_i\|+\prod\limits_{i = 1}^D \|x_i\|, \; x_i \in[-10, 10]$	$\begin{array}{l} x_i = 0, \\ f({{{\mathit{\boldsymbol{X}}}}}) = 0 \\ \end{array}$	Unimodal
$f_3 $ Rastrigin	$f({{{\mathit{\boldsymbol{X}}}}}) = { }\sum\limits_{i = 1}^D (100(x_i^2 -10\cos (2{\rm{\pi }} x_i)+10)), \; x_i \in [-5.12, 5.12]$	$\begin{array}{l} x_i = 0, \\ f({{{\mathit{\boldsymbol{X}}}}}) = 0 \\ \end{array}$	Multi-modal
$f_4 $ Griewank	$f({{{\mathit{\boldsymbol{X}}}}}) = { }\frac{1}{4000}\sum\limits_{i = 1}^D {x_i^2 -\prod\limits_{i = 1}^D {\cos\frac{x_i }{\sqrt i }+1} }, \; x_i \in [-600, 600]$	$\begin{array}{l} x_i = 0, \\ f({{{\mathit{\boldsymbol{X}}}}}) = 0 \\ \end{array}$	Multi-modal
$f_5 $ Ackley	$\begin{array}{l} f({{{\mathit{\boldsymbol{X}}}}}) = -20{ }\exp \left(-0.2\sqrt {\frac{1}{n}\sum\limits_{i = 1}^D {x_i^2 } } \right)- \\ \exp { }\left(\frac{1}{n}\sum\limits_{i = 1}^D {\cos (2{\rm{\pi }} x_i)+20+{{{\rm{e}}}}} \right), \; x_i \in [-32, 32] \\ \end{array}$	$\begin{array}{l} x_i = 0, \\ f({{{\mathit{\boldsymbol{X}}}}}) = 0 \\ \end{array}$	Multi-modal
$f_6 $ Schwefel	$f({{{\mathit{\boldsymbol{X}}}}}) = -{ }\sum\limits_{i = 1}^D(x_i \sin \sqrt {\|x_i\|}), \; x_i \in [-500, 500]$	$\begin{array}{l} x_i = 420.968\; 7, \\ f({{{\mathit{\boldsymbol{X}}}}}) = -418.982\; 9D \\ \end{array}$	Multi-modal
$f_7 $	$f({{{\mathit{\boldsymbol{X}}}}}) = { }\sum\limits_{i = 1}^D\left(\sum\limits_{j-1}^i {x_j } \right)^2, \; x_i \in [-100, 100]$	$\begin{array}{l} x_i = 0, \\ f({{{\mathit{\boldsymbol{X}}}}}) = 0 \\ \end{array}$	Unimodal
$f_8 $	$f({{{\mathit{\boldsymbol{X}}}}}) = { }\max \limits_i \{\|x_i\|, 1{\leqslant} i{\leqslant} D\}, \; x_i \in [-100, 100]$	$\begin{array}{l} x_i = 0, \\ f({{{\mathit{\boldsymbol{X}}}}}) = 0 \\ \end{array}$	Unimodal
$f_9 $	$\begin{array}{l} f({{{\mathit{\boldsymbol{X}}}}}) = { }\sum\limits_{i = 1}^{D-1} {[100(x_{i+1}-x_i ^2)^2+(x_i-1)^2], } \\ x_i \in [-30, 30] \\ \end{array}$	$\begin{array}{l} x_i = 0, \\ f({{{\mathit{\boldsymbol{X}}}}}) = 0 \\ \end{array}$	Unimodal
$f_{10} $	$f({{{\mathit{\boldsymbol{X}}}}}) = { }\sum\limits_{i = 1}^D {([x_i +0.5])^2}, \; x_i \in[-100, 100]$	$\begin{array}{l} x_i = 0, \\ f({{{\mathit{\boldsymbol{X}}}}}) = 0 \\ \end{array}$	Unimodal
$f_{11} $	$f(x) = { }\sum\limits_{i = 1}^D {ix_i^4 +{{{\rm{random}}}}} [0, 1), \; x_i \in[-1.28, 1.28]$	$\begin{array}{l} x_i = 0, \\ f({{{\mathit{\boldsymbol{X}}}}}) = 0 \\ \end{array}$	Unimodal
$f_{12} $	$\begin{array}{l} f({{{\mathit{\boldsymbol{X}}}}}) = { }\frac{{\rm{\pi }} }{D}\{10\sin ({\rm{\pi }} y_1)+ \\ { }\sum\limits_i^{D-1} {(y_i -1)^2[1+10\sin ^2({\rm{\pi }} y_{i+1})]+(y_D -1)^2} \} +\\ { }\sum\limits_{i = 1}^D {u(x_i, 10, 100, 4)}, \; y_i = 1+\frac{x_i +1}{4} \\ u(x_i, a, k, m) = \left\{\!\!\begin{array}{l} k(x_i -a)^mx_i > a \\ 0-a < x_i < a \\ k(-x_i -a)^mx_i < -a \\ \end{array}\!\!\right.\ \end{array}$	$\begin{array}{l} x_i = 0, \\ f({{{\mathit{\boldsymbol{X}}}}}) = 0 \\ \end{array}$	Multi-modal
$f_{13} $	$\begin{array}{l} f({{{\mathit{\boldsymbol{X}}}}}) = 0.1\{\sin ^2(3{\rm{\pi }} x_1)+ \\ { }\sum\limits_{i = 1}^D {(x_i -1)^2[1+\sin ^2(3{\rm{\pi }} x_i +1)]} +(x_D -1)^2[1+\sin ^2(2{\rm{\pi }} x_D)]\}+ \\ { }\sum\limits_{i = 1}^D {u(x_i, 5, 100, 4)}, \; x_i \in [-50, 50] \\ \end{array}$	$\begin{array}{l} x_i = 0, \\ f({{{\mathit{\boldsymbol{X}}}}}) = 0 \\ \end{array}$	Multi-modal

Table 1

Fig 1

Fig 2

Fig 3

Fig 4

Fig 5

Fig 6

Table 2

Optimization results comparison of benchmark functions"

Function	Dimension	Algorithm	Average	Variance	Maximum	Minimum	Convergence time/s
$f_1 $	50	SATC-ALO	0	0	0	0	1.2
		ALO	3.97e-11	2.56e-11	4.24e-11	3.68e-11	183.2
		GWO	5.46e-17	3.71e-05	6.28e-17	4.87e-17	194.7
		PSO	4.16e-08	6.53e-08	5.01e-08	3.23e-08	207.6
		ABC	5.75e-09	3.46e-09	6.39e-09	4.14e-09	203.4
	30	SATC-ALO	0	0	0	0	0.6
		ALO	5.32e-12	3.14e-12	6.28e-12	4.37e-12	63.8
		GWO	6.57e-20	6.12e-05	7.26e-20	4.99e-20	62.8
		PSO	8.26e-10	5.13e-10	9.17e-10	7.35e-10	145.3
		ABC	6.82e-11	4.66e-11	7.95e-11	5.24e-11	118.7
$f_2 $	50	SATC-ALO	0	0	0	0	1.1
		ALO	5.78e-06	7.65e-07	6.24e-06	4.12e-06	184.5
		GWO	5.31e-13	4.82e-03	6.55e-13	3.95e-13	190.2
		PSO	3.76e-04	6.73e-04	5.34e-04	2.41e-04	204.8
		ABC	4.92e-04	5.42e-04	6.42e-04	3.22e-04	206.2
	30	SATC-ALO	0	0	0	0	0.1
		ALO	6.92e-07	2.68e-07	8.34e-07	5.31e-07	65.1
		GWO	7.31e-15	5.48e-04	9.52e-15	5.66e-15	71.3
		PSO	5.84e-05	6.73e-05	6.88e-05	4.63e-05	94.0
		ABC	3.92e-05	5.86e-05	4.36e-05	2.85e-05	89.6
$f_3 $	50	SATC-ALO	0	0	0	0	1.7
		ALO	7.26e-07	5.34e-08	9.19e-07	6.28e-07	156.6
		GWO	8.24	10.09	10.52	7.46	161.3
		PSO	69.32	43.96	75.69	60.22	180.4
		ABC	27.13	20.18	35.97	19.28	178.5
	30	SATC-ALO	0	0	0	0	0.5
		ALO	8.46e-08	1.49e-08	10.61e-08	6.82e-08	63.8
		GWO	0.31	0.49	1.69	0.04	65.3
		PSO	25.86	11.37	31.84	22.85	72.9
		ABC	1.79	2.85	2.75	0.93	70.8
$f_4 $	50	SATC-ALO	0	0	0	0	1.3
		ALO	2.31e-02	5.18e-02	4.28e-02	8.7e-03	180.8
		GWO	7.21e-02	4.31e-02	8.95e-02	5.54e-02	187.6
		PSO	6.54	3.73	8.63	5.47	201.7
		ABC	5.81	2.17	6.85	4.74	197.2
	30	SATC-ALO	0	0	0	0	0.5
		ALO	6.54e-03	2.79e-04	9.64e-03	4.17e-03	55.9
		GWO	4.49e-03	3.62e-03	6.57e-03	4.22e-03	53.4
		PSO	0.27	0.21	1.21	0.01	60.6
		ABC	0.73	0.49	2.58	0.02	59.3
$f_5 $	50	SATC-ALO	8.88e-16	8.59e-17	10.83e-16	7.21e-16	3.3
		ALO	7.34e-09	5.94e-10	9.84e-09	5.78e-09	179.8
		GWO	3.76e-08	6.43e-08	5.65e-08	1.21e-08	182.9
		PSO	7.06e-05	3.61e-05	9.36e-05	5.24e-05	195.4
		ABC	5.72e-06	4.76e-06	6.85e-06	3.55e-06	193.6
	30	SATC-ALO	0	0	0	0	1.3
		ALO	5.71e-15	2.48e-15	7.42e-15	4.13e-15	63.3
		GWO	1.06e-13	0.78e-13	1.25e-13	0.43e-13	75.7
		PSO	1.11e-09	2.39e-09	2.55e-09	0.24e-09	91.6
		ABC	2.88e-10	8.16e-10	4.22e-10	1.68e-10	89.4
$f_6 $	50	SATC-ALO	–2.09e+04	1.32e+04	–1.38e+04	–3.57e+04	3.3
		ALO	–8.07e+03	7.35e+03	–7.85e+03	–9.07e+03	181.6
		GWO	–6.60e+03	3.59e+03	–4.85e+03	–8.63e+03	187.9
		PSO	–4.78e+03	2.38e+03	–3.58e+03	–5.25e+03	194.5
		ABC	–5.03e+03	2.51e+03	–3.59e+03	–6.11e+03	192.0
	30	SATC-ALO	–8.44e+03	7.19e+02	–8.36e+03	–9.85e+03	2.1
		ALO	–6.39e+03	3.18e+02	–6.21e+03	–7.96e+03	68.2
		GWO	–6.85e+03	3.86e+02	–5.88e+03	–8.41e+03	70.6
		PSO	–2.27e+03	1.35e+02	–1.87e+03	–4.87e+03	79.5
		ABC	–3.18e+03	1.78e+02	–1.79e+03	–4.88e+03	80.3
$f_7 $	50	SATC-ALO	0	0	0	0	3.9
		ALO	2.63e-07	5.34e-07	3.25e-07	2.26e-07	159.7
		GWO	1.39e-04	7.62e-03	2.71e-04	1.03e-04	162.2
		PSO	6.08e-03	4.39e-03	6.94e-03	4.65e-03	185.3
		ABC	5.91e-03	6.26e-03	8.66e-03	1.58e-03	184.7
	30	SATC-ALO	0	0	0	0	1.2
		ALO	6.69e-10	6.34e-10	6.97e-10	5.41e-10	62.3
		GWO	3.23e-06	7.91e-06	4.36e-06	2.87e-06	67.7
		PSO	4.53e-06	1.47e-06	4.74e-06	3.52e-06	80.5
		ABC	5.48e-06	3.72e-06	6.35e-06	4.55e-06	71.5
$f_8 $	50	SATC-ALO	4.46e-16	1.28e-16	5.88e-16	4.25e-16	4.3
		ALO	5.46e-06	3.22e-06	6.25e-06	3.28e-06	193.4
		GWO	6.29e-05	4.37e-05	8.36e-05	4.25e-05	197.9
		PSO	9.17	4.38	10.87	5.66	218.0
		ABC	2.19e-02	3.44e-02	5.35e-02	1.96 e-02	206.5
	30	SATC-ALO	0	0	0	0	2.0
		ALO	1.36e-08	1.81e-09	2.99e-08	5.10e-09	165.4
		GWO	5.61e-07	3.76e-08	7.69e-07	4.25e-07	160.2
		PSO	1.09	0.32	2.52	0.85	185.3
		ABC	3.73e-04	2.19e-04	4.33e-04	1.96e-04	179.6
$f_9 $	50	SATC-ALO	2.41e-02	6.67e-02	3.36e-02	1.55e-02	7.4
		ALO	8.93	11.43	13.85	7.58	261.9
		GWO	44.86	60.21	50.96	35.74	271..4
		PSO	117.6	83.23	125.96	100.36	293.5
		ABC	63.85	45.79	69.66	60.52	284.6.
	30	SATC-ALO	4.67e-03	3.77e-02	4.95e-03	3.74e-03	4.2
		ALO	0.35	0.11	0.75	0.21	91.5
		GWO	26.81	69.90	27.66	25.79	94.7
		PSO	96.71	60.11	99.74	95.85	136.3
		ABC	46.97	35.56	48.66	39.97	128.5
$f_{10} $	50	SATC-ALO	3.98e-15	2.14e-15	4.54e-15	2.87e-15	3.4
		ALO	1.12e-09	8.16e-09	2.21e-09	9.81e-10	126.9
		GWO	0.51	1.37	1.83	0.29	132.3
		PSO	2.76e-05	7.79e-05	3.44e-05	1.58e-05	157.8
		ABC	4.92e-06	3.96e-06	5.77e-06	3.85e-06	153.4
	30	SATC-ALO	0	0	0	0	1.5
		ALO	2.56e-10	1.09e-10	3.14e-10	1.19e-10	53.4
		GWO	7.48e-06	3.76e-06	7.88e-06	6.97e-06	60.3
		PSO	6.14e-07	2.86e-06	6.58e-07	5.47e-07	82.5
		ABC	4.81e-06	8.25e-06	5.11e-06	2.98e-06	81.2
$f_{11} $	50	SATC-ALO	7.17e-06	1.41e-06	8.58e-06	5.79e-06	5.8
		ALO	4.16e-03	3.62e-03	4.65e-03	2.99e-03	183.7
		GWO	7.35e-02	4.88e-02	8.87e-02	6.12e-02	186.4
		PSO	8.63	5.85	9.98	7.28	198.5
		ABC	0.79	0.23	1.41	9.8e-02	194.7
	30	SATC-ALO	4.89e-10	4.17e-10	5.25e-10	3.89e-10	2.9
		ALO	3.23e-04	8.59e-04	4.58e-04	2.98e-04	84.9
		GWO	2.21e-03	5.52e-03	3.25e-03	1.58e-03	87.5
		PSO	0.12	1.59	1.25	9.4e-02	102.4
		ABC	2.06e-02	4.25e-02	2.88e-02	1.97e-02	96.5
$f_{12} $	50	SATC-ALO	0	0	0	0	3.6
		ALO	6.54e-08	3.86e-08	9.25e-08	5.14e-08	165.7
		GWO	3.81e-02	1.31e-02	4.62e-02	2.44e-02	173.9
		PSO	4.47e-05	2.58e-05	4.97e-05	3.47e-05	184.8
		ABC	1.03e-06	7.24e-06	2.86e-06	0.75e-06	182.5
	30	SATC-ALO	0	0	0	0	1.9
		ALO	2.65e-12	5.49e-12	3.09e-12	2.11e-12	68.3
		GWO	9.51e-04	5.63e-04	11.74e-04	8.14e-04	72.4
		PSO	8.46e-09	7.35e-09	9.87e-09	7.84e-09	85.9
		ABC	6.84e-07	2.44e-07	7.41e-07	5.11e-07	83.6
$f_{13} $	50	SATC-ALO	0	0	0	0	3.2
		ALO	1.18e-11	5.06e-11	2.25e-11	1.02e-11	149.0
		GWO	3.81e-02	1.31e-02	3.98e-02	2.98e-02	156.3
		PSO	4.63e-05	1.24e-05	5.37e-05	4.24e-05	169.4
		ABC	6.09e-06	2.88e-06	6.25e-06	5.76e-06	167.2
	30	SATC-ALO	0	0	0	0	1.4
		ALO	2.65e-12	5.49e-12	3.54e-12	1.15e-12	57.7
		GWO	4.92e-06	2.25e-06	3.87e-06	2.14e-06	61.5
		PSO	3.78e-07	8.28e-07	4.54e-07	2.98e-07	68.9
		ABC	5.63e-08	4.88e-08	6.47e-08	4.15e-08	67.2

Table 2

Fig 7

Fig 8

Fig 9

Fig 10

Table 3

Fig 11

Fig 12

Table 4

Fig 13

Fig 14

Table 5

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Position of threats/km	Radius of threats/km
[20, 25]	10
[25, 35]	16
[48, 27]	8
[50, 40]	16
[70, 65]	10
[75, 50]	14
[90, 15]	10
[80, 70]	10

Position of threats/km	Radius of threats/km
[85, 25]	15
[45, 60]	10
[80, 63]	13
[100, 40]	4
[45, 35]	8

Algorithm	Path cost
Algorithm	Average	Maximum	Minimum
SATC-ALO	331.69	340.27	322.64
ALO	378.77	386.31	370.52
GWO	384.26	390.58	375.47
PSO	400.73	410.94	393.25
ABC	390.65	398.63	382.83

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Antlion optimizer algorithm based on chaos search and its application

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