Improved Cuckoo Search algorithm for numerical function optimization

Jianjun Liu; Min Zeng; Yifan Ge; Changzhi Wu; Xiangyu Wang

doi:10.3934/jimo.2018142

Article Contents

2020, Volume 16, Issue 1: 103-115. Doi: 10.3934/jimo.2018142

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Improved Cuckoo Search algorithm for numerical function optimization

1.
School of Science, China University of Petroleum, Beijing 102249, China
2.
School of Management, Guangzhou University, Guangzhou, 510006, China
3.
Australasian Joint Research Centre for Building Information Modelling, School of Built Environment, Curtin University, 6102, WA, Australia

* Corresponding author: Changzhi Wu
* Corresponding author: Changzhi Wu

Received: February 2016

Revised: January 2018

Early access: September 2018

Published: October 2019

This paper is partially supported by the National Natural Science Foundation of China (61473326) and Australian Research Council.

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Abstract

Cuckoo Search (CS) is a recently proposed metaheuristic algorithm to solve optimization problems. For improving its performance both on the efficiency of searching and the speed of convergence, we proposed an improved Cuckoo Search algorithm based on the teaching-learning strategy (TLCS). For a better balance between intensification and diversification, both a dynamic weight factor and an out-of-bound project strategies are also introduced into TLCS. The results of numerical experiment demonstrate that our improved TLCS performs better than the basic CS and other two improved CS methods appearing in literatures.

Keywords:

Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.

Citation:

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Figure 1. Project plot. $x_1, x_2, x_3$ are transformed to $y_1, y_2, y_3$, respectively, by (11) while $x_4$ is first transformed to $y_4$ by (11), then to $y_4'$ by (12).

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Figure 2. Convergence performance of CS, ASCS, WCS and TLCS, on 8 test functions of 5 dimensions. Horizontal and vertical axes represent the number of iterations and the logarithm of the function values, respectively

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Table 1. 18 benchmark problems of $n(\geq 2)$ dimensions used in experiments. $x = (x_1, x_2, \cdots, x_n)$, C: characteristic, Opt.: optimal function value, U.: unimodal, M.: multimodal, S.: separable, N.: non-separable.

Function Name	C	Range	Formula	Opt.
Sphere	S.U.	[-100,100]	$f_1(x)= \sum\limits_{i=1}^{n}x_{i}^2$	0
Ackley	S.M.	[-32, 32]	$ f_2(x)=20-20 e^{-0.2\sqrt{\frac{1}{n}\sum\limits_{i=1}^n x_i^2}} -e^{\frac{1}{n}\sum\limits_{i=1}^n \cos (2\pi x_i)} +e$	0
Sum-Squares	S.U.	$[-10, 10]$	$f_{3}(x)=\sum\limits_{i=1}^n ix_i^2$	0
Rastrigin	S.M.	[-5.12, 5.12]	$f_4(x) = 10n + \sum\limits_{i=1}^n \big[x_i^2 -10\cos(2\pi x_i)\big]$	0
Rosenbrock	N.U.	[-10, 10]	$f_5(x)= \sum\limits_{i=1}^{n-1}\big[100(x_{i}^2 - x_{i+1})^2 + (x_i -1)^2\big]$	0
Griewank	N.M.	[-600,600]	$f_{6}(x) = 1 + \frac{1}{4000} \sum\limits_{i=1}^n x^2_i - \prod\limits_{i=1}^n \cos(\frac{x_i}{\sqrt{i}})$	0
Levy	N.M.	[-10, 10]	$f_{7}(x)={\sin(\pi w_1)}^2 +\sum\limits_{i=1}^{n-1}\big \{{(w_i-1)}^2\big[1+10$
			${\sin(\pi w_i+1)}^2\big]\big\}+{(w_n-1)}^2[1+{\sin(2\pi w_n)]}^2$,	0
			where $w_i=1+\frac{x_i-1}{4}$, for all $i=1, 2, \cdots, n$
Powell	N.M	[-4, 5]	$f_{8}(x)=\sum\limits_{i=1}^\frac{n}{4} \Big[{(x_{4i-3}+10x_{4i-2})}^2+ 5(x_{4i-1}-x_{4i-2})^2$	0
			$ + (x_{4i-2}-2x_{4i-1})^4+ 10(x_{4i-3}-x_{4i})^4\Big]$
Zakharov	N.-	[-5, 10]	$f_9(x)=\sum\limits_{i=1}^{n}x_{i}^2 + \Big(\frac{1}{2}\sum\limits_{i=1}^{n}ix_{i}\Big)^2+\Big( \frac{1}{2}\sum\limits_{i=1}^{n}ix_{i}\Big)^4$	0
Weierstrass	N.M.	$[-0.5, 0.5]$	$f_{10}(x)=\sum\limits_{i=1}^n\big(\sum_{k=0}^{k_{max}}[a^k\cos(2\pi b^k(x_i+0.5))]\big)$	0
			$~~-n\sum_{k=0}^{k_{max}}[a^k\cos(2\pi b^k\cdot0.5)], $
			$a=0.5, b=3, k_{max}=20$
Schwefel	S.M.	[-500,500]	$f_{11}(x) = 418.982887 \cdot n-\sum\limits_{i=1}^n \big(x_i \sin(\sqrt{\|x_i\|}\big)$	0
Whitley	N.M.	[-10.2, 10.2]	$f_{12}(x) = \sum\limits_{i=1}^n \sum\limits_{j=1}^n \Big[\frac{(100(x_i^2-x_j)^2 + (1-x_j)^2)^2}{4000} $	0
			$-\cos\big(100(x_i^2-x_j)^2 + (1-x_j)^2\big)+1\Big]$
Trid	N.-	$[-n^2, n^2]$	$f_{13}(x)=\sum\limits_{i=1}^{n}(x_i-1)^2- \sum\limits_{i=1}^{n}x_ix_{i-1}, (n=5)$	$-30$
			$(n=10)$	$-200$
Styblinski-Tang	S.M.	$[-5, 5]$	$f_{14}(x)=\frac{1}{n}\sum\limits_{k=1}^n (x_i^4-16x_i^2+5x_i), a=39.16599$	$-an$
Alpine	N.M.	$[-10, 10]$	$f_{15}(x)=\sum\limits_{i=1}^n \|x_i\cdot \sin(x_i)+0.1\cdot x_i\|$	0
Easom	S.U.	$[-100,100]$	$f_{16}(x)=-\cos(x_1)\cos(x_2)e^{-(x_1-pi)^2-(x_2-pi)^2}$	-1
Treccani	S.M.	$[-3, 3]$	$f_{17}(x)=(x_1)^4+4(x_1)^3+4(x_1)^2+(x_2)^2$	0
Perm$(n, \beta)$	N.U	$[-n, n]$	$f_{18}(x)=\sum\limits_{k=1}^n {\Big[\sum\limits_{j=1}^n (j^k+\beta) \left({x_j^ k-\frac{1}{j^k}}\right)\Big]}^2, \beta=0.5$	0

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Table 2. Statistic results obtained by CS [24], ASCS [25], WCS [28] and TLCS on 5-D functions over 30 independent runs, respectively

Fun.	Indictor	CS	ASCS	WCS	TLCS
$f_{1}$	Best	4.5213e-14	2.4536e-18	0.0000e+00	0.0000e+00
	Mean	2.6910e-12	1.7970e-16	0.0000e+00	0.0000e+00
	Std.	4.2657e-12	2.0513e-16	0.0000e+00	0.0000e+00
$f_{2}$	Best	5.3705e-05	9.3072e-05	8.8818e-16	8.8818e-16
	Mean	3.7553e-04	6.6183e-04	8.8818e-16	8.8818e-16
	Std.	4.3639e-04	6.6652e-04	0.0000e+00	0.0000e+00
$f_{3}$	Best	9.9825e-16	5.7419e-19	0.0000e+00	0.0000e+00
	Mean	3.5974e-14	1.8282e-17	0.0000e+00	0.0000e+00
	Std.	4.4513e-14	2.9784e-17	0.0000e+00	0.0000e+00
$f_{4}$	Best	2.1760e-02	6.0844e-02	0.0000e+00	0.0000e+00
	Mean	4.1580e-01	6.0469e-01	0.0000e+00	0.0000e+00
	Std.	4.1771e-01	4.3998e-01	0.0000e+00	0.0000e+00
$f_{5}$	Best	5.4423e-04	3.8454e-03	9.0786e-05	3.2173e-06
	Mean	4.9548e-02	8.8289e-02	4.9515e-03	9.0026e-05
	Std.	8.2985e-02	1.2769e-01	4.4433e-03	2.4935e-04
$f_{6}$	Best	2.4318e-02	1.3526e-02	0.0000e+00	0.0000e+00
	Mean	4.0590e-02	4.9268e-02	1.6139e-02	0.0000e+00
	Std.	1.1239e-02	2.1748e-02	2.6449e-02	0.0000e+00
$f_{7}$	Best	7.4053e-12	4.1549e-11	1.4224e-13	6.0523e-17
	Mean	4.5672e-10	5.7734e-10	4.9219e-11	1.3148e-15
	Std.	1.2514e-09	5.2112e-10	4.1785e-11	2.0139e-15
$f_{8}$	Best	7.5663e-19	1.9396e-20	0.0000e+00	5.2592e-51
	Mean	3.2565e-17	1.1242e-17	0.0000e+00	1.0280e-44
	Std.	4.6518e-17	1.5804e-17	0.0000e+00	2.1851e-44
$f_{9}$	Best	2.7592e-13	1.2952e-16	0.0000e+00	0.0000e+00
	Mean	2.6449e-12	1.6090e-15	0.0000e+00	0.0000e+00
	Std.	2.9165e-12	1.4999e-15	0.0000e+00	0.0000e+00
$f_{10}$	Best	5.5995e-03	6.6042e-03	0.0000e+00	0.0000e+00
	Mean	1.1684e-02	1.0426e-02	0.0000e+00	0.0000e+00
	Std.	6.3211e-03	3.5660e-03	0.0000e+00	0.0000e+00
$f_{11}$	Best	2.0752e+03	2.0752e+03	2.0752e+03	2.0225e+03
	Mean	2.0752e+03	2.0752e+03	2.0752e+03	2.0616e+03
	Std.	4.7941e-11	1.7728e-10	1.5453e-09	2.2077e+01
$f_{12}$	Best	1.1195e-03	1.3950e-03	1.1187e-02	9.8810e-13
	Mean	7.2247e-01	5.6108e-01	1.5719e+00	5.6092e+00
	Std.	9.9970e-01	8.9282e-01	1.2832e+00	4.2152e+00
$f_{13}$	Best	-3.0000e+01	-3.0000e+01	-3.0000e+01	-3.0000e+01
	Mean	-3.0000e+01	-3.0000e+01	-3.0000e+01	-3.0000e+01
	Std.	2.3597e-13	0.0000e+00	4.2507e-12	6.6486e-14
$f_{14}$	Best	-7.8332e+01	-7.8332e+01	-7.8332e+01	-7.8332e+01
	Mean	-7.8332e+01	-7.8332e+01	-7.8332e+01	-7.7201e+01
	Std.	5.0220e-07	8.9447e-07	1.2933e-06	2.3413e+00
$f_{15}$	Best	1.7330e-03	3.1350e-03	1.1926e-181	0.0000e+00
	Mean	8.1827e-03	1.4765e-02	3.9923e-177	0.0000e+00
	Std.	4.3095e-03	1.2199e-02	0.0000e+00	0.0000e+00
$f_{16}$	Best	-1.0000e+00	-1.0000e+00	-1.0000e+00	-1.0000e+00
	Mean	-1.0000e+00	-1.0000e+00	-1.0000e+00	-1.0000e+00
	Std.	1.0624e-13	1.3743e-13	0.0000e+00	0.0000e+00
$f_{17}$	Best	-3.5527e-15	-3.5527e-15	0.0000e+00	0.0000e+00
	Mean	-3.3892e-15	-3.1925e-15	0.0000e+00	2.0292e-67
	Std.	6.3290e-16	9.9925e-16	0.0000e+00	7.8473e-67
$f_{18}$	Best	2.5826e-05	5.0852e-04	3.9689e-07	1.0547e-13
	Mean	6.0659e-02	6.3264e-02	1.7993e-05	4.1372e-12
	Std.	1.2079e-01	1.0323e-01	1.8082e-05	4.4431e-12

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Table 3. Statistic results obtained by CS [24], ASCS [25], WCS [28] and TLCS on 10-D functions over 30 independent runs, respectively

Fun.	Indictor	CS	ASCS	WCS	TLCS
$f_{1}$	Best	3.2389e-06	1.1207e-08	0.0000e+00	0.0000e+00
	Mean	8.0057e-06	5.3375e-08	0.0000e+00	0.0000e+00
	Std.	4.6479e-06	2.9549e-08	0.0000e+00	0.0000e+00
$f_{2}$	Best	3.7125e-02	8.7783e-02	8.8818e-16	8.8818e-16
	Mean	6.0272e-01	4.0848e-01	8.8818e-16	8.8818e-16
	Std.	5.8084e-01	2.2043e-01	0.0000e+00	0.0000e+00
$f_{3}$	Best	1.8590e-07	5.7623e-10	0.0000e+00	0.0000e+00
	Mean	5.2596e-07	2.7587e-09	0.0000e+00	0.0000e+00
	Std.	2.9285e-07	2.3201e-09	0.0000e+00	0.0000e+00
$f_{4}$	Best	6.0868e+00	9.9616e+00	0.0000e+00	0.0000e+00
	Mean	1.1568e+01	1.5915e+01	0.0000e+00	0.0000e+00
	Std.	2.4106e+00	4.5746e+00	0.0000e+00	0.0000e+00
$f_{5}$	Best	5.5106e-01	1.1646e+00	2.7452e+00	2.2219e+00
	Mean	4.3244e+00	3.2423e+00	3.6089e+00	2.7140e+00
	Std.	1.4901e+00	1.5008e+00	3.7702e-01	3.3872e-01
$f_{6}$	Best	5.4516e-02	4.9131e-02	0.0000e+00	0.0000e+00
	Mean	7.6544e-02	9.0650e-02	0.0000e+00	0.0000e+00
	Std.	1.4500e-02	2.1017e-02	0.0000e+00	0.0000e+00
$f_{7}$	Best	1.6338e-04	3.3752e-03	4.5958e-06	9.2416e-10
	Mean	1.0474e-02	2.5951e-02	1.2202e-04	1.7004e-08
	Std.	1.0911e-02	1.8264e-02	1.1195e-04	1.4461e-08
$f_{8}$	Best	6.3186e-09	1.5939e-09	0.0000e+00	2.1348e-36
	Mean	4.4004e-08	9.4253e-09	0.0000e+00	7.5924e-26
	Std.	4.4362e-08	7.1640e-09	0.0000e+00	2.3933e-25
$f_{9}$	Best	8.7140e-04	3.7275e-05	0.0000e+00	0.0000e+00
	Mean	4.8839e-03	8.7891e-05	0.0000e+00	0.0000e+00
	Std.	3.2252e-03	4.7437e-05	0.0000e+00	0.0000e+00
$f_{10}$	Best	4.0273e-01	4.6694e-01	0.0000e+00	0.0000e+00
	Mean	7.8045e-01	8.4527e-01	0.0000e+00	0.0000e+00
	Std.	2.4665e-01	2.5778e-01	0.0000e+00	0.0000e+00
$f_{11}$	Best	4.1504e+03	4.1504e+03	4.1504e+03	4.0458e+03
	Mean	4.1504e+03	4.1504e+03	4.1504e+03	4.1152e+03
	Std.	1.1924e-04	6.4787e-04	2.7421e-04	3.9471e+01
$f_{12}$	Best	2.7106e+01	3.3064e+01	2.4671e+01	4.0486e+01
	Mean	4.2321e+01	4.4884e+01	3.8730e+01	4.4356e+01
	Std.	6.7034e+00	6.0260e+00	4.9097e+00	1.7008e+00
$f_{13}$	Best	-1.2467e+02	-1.2467e+02	-1.2467e+02	-1.7996e+02
	Mean	-1.2467e+02	-1.2467e+02	-1.2467e+02	-1.6700e+02
	Std.	6.0730e-09	5.1284e-10	4.6884e-08	6.7526e+00
$f_{14}$	Best	-7.8281e+01	-7.8225e+01	-7.8051e+01	-7.8332e+01
	Mean	-7.7819e+01	-7.6712e+01	-7.6642e+01	-7.6256e+01
	Std.	3.5470e-01	1.4800e+00	1.1564e+00	2.9181e+00
$f_{15}$	Best	2.2444e-01	5.7034e-01	2.5015e-181	0.0000e+00
	Mean	3.8266e-01	1.1298e+00	4.8905e-179	0.0000e+00
	Std.	1.0949e-01	4.0051e-01	0.0000e+00	0.0000e+00
$f_{16}$	Best	-1.0000e+00	-1.0000e+00	-1.0000e+00	-1.0000e+00
	Mean	-1.0000e+00	-1.0000e+00	-1.0000e+00	-1.0000e+00
	Std.	9.4746e-13	6.1618e-13	0.0000e+00	0.0000e+00
$f_{17}$	Best	-3.5527e-15	-3.5527e-15	0.0000e+00	0.0000e+00
	Mean	-3.3158e-15	-3.3945e-15	0.0000e+00	2.7979e-35
	Std.	9.1735e-16	5.0210e-16	0.0000e+00	1.0836e-34
$f_{18}$	Best	1.0000e+10	1.0000e+10	3.2642e+01	2.4003e-05
	Mean	1.0000e+10	1.0000e+10	6.0712e+01	5.3966e-04
	Std.	0.0000e+00	0.0000e+00	2.0484e+01	4.9058e-04

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Table 4. Wilcoxon rank sum results on best and average function values obtained by TLCS against CS, ASCS and WCS algorithms for 18 benchmark functions

Algorithm	Dim.	criteria	rank sum	$p$ value	$h$ value
TLCS to CS		Best	421.5000	0.0051	1
		Mean	411.0000	0.0139	1
		Std	422.0000	0.0048	1
TLCS to ASCS		Best	415.0000	0.0095	1
	5	Mean	406.0000	0.0213	1
		Std	408.5000	0.0165	1
TLCS to WCS		Best	357.0000	0.4363	0
		Mean	356.5000	0.4532	0
		Std	333.0000	1.0000	0
TLCS to CS		Best	410.5000	0.0143	1
		Mean	407.0000	0.0196	1
		Std	405.5000	0.0213	1
TLCS to ASCS		Best	408.5000	0.0170	1
	10	Mean	406.0000	0.0213	1
		Std	407.5000	0.0180	1
TLCS to WCS		Best	353.0000	0.5183	0
		Mean	348.0000	0.6339	0
		Std	324.5000	0.7810	0

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