Article Contents
Article Contents

Improved Cuckoo Search algorithm for numerical function optimization

• * Corresponding author: Changzhi Wu
This paper is partially supported by the National Natural Science Foundation of China (61473326) and Australian Research Council.
• 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.

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

 Citation:

• 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).

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

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

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

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

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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