# K-Means Clustering Algorithm Based on Chaotic Adaptive Artificial Bee Colony
## 4 Numerical Experiments
### CAABC Performance Analysis  
To demonstrate the superiority and effectiveness of the CAABC, the CAABC algorithm is compared with other well-known algorithms on twenty benchmark problems. The population parameter settings are same as the setting mentioned in [36]: \(N=25\), the maximum number of evaluations \(\max\)\(cycle=10,000*D\), and other function parameter settings are shown in Table 2. Tables 3 and 4 demonstrate the comparison under the 30-dimensional and 60-dimensional parameter settings respectively. The best results are shown in bold. All algorithms are executed in the same machine environment. Each \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Number** & **Equation** & **Name** & **Domain** \\ \hline
7 & \(f_{0}\varpi(x)=\sum\limits_{i=1}^{D}{z_{i}^{2}z=x-}\) & ShiftedSphere & \([-100,100]^{D}\) \\ \hline
8 & \(f_{08}(x)=\sum\limits_{i=1}^{D}{(x_{i}+0.5)^{2}}\) & Step & \([-100,100]^{D}\) \\ \hline
9 & \(f_{9}=\sum\limits_{i=1}^{D}{x_{i}^{2}+\left(\sum\limits_{i=1}^{D}{0.5 ix_{i}} \right)^{2}+\left(\sum\limits_{i=1}^{D}{0.5 ix_{i}}\right)^{4}}\) & Zakharow & \([-5,5]^{D}\) \\ \hline
10 & \(f_{10}(x)=\sum\limits_{i=1}^{D}{ix_{i}^{2}}\) & SumQuares & \([-10,10]^{D}\) \\ \hline
11 & \(f_{11}(x)=\sum\limits_{i=1}^{D}{[x_{i}]^{(i+1)}}\) & SumDifference & \([-10,10]^{D}\) \\ \hline
12 & \(f_{12}(x)=-418.98288727243369\times D\) & Schwefel2.26 & \([-500,500]^{D}\) \\  & \(+\sum\limits_{i=1}^{D}{[-x_{i}\sin(x_{i})\sqrt{{\left|{x_{i}}\right|}}]}\) & Schwefel2.26 & \([-500,500]^{D}\) \\ \hline
13 & \(f_{13}(x)=\sum\limits_{i=1}^{D}{[100(z_{i+1}-x_{i}^{2})^{2}+\left(z_{i}-1 \right)^{2}]}\), & ShiftedRosenbrock & \([-10,10]^{D}\) \\ \hline
14 & \(f_{14}(x)=\sum\limits_{i=1}^{D}{\left(\sum\limits_{i+1}^{i}{x_{i}}\right)^{2}}\) & Schwefel1.2 & \([-100,100]^{D}\) \\ \hline
15 & \(f_{15}=-20 exp\left(-0.2\sqrt{\frac{1}{D}\sum\limits_{i=1}^{D}{x_{i}^{2}}}\right)\) & Ackley & \([-32,32]^{D}\) \\ \hline
16 & \(f_{16}(x)=\sum\limits_{i=1}^{D}{(x_{i}^{2}-10 cos(2\pi x)+10)}\) & Griewank & \([-600,600]^{D}\) \\ \hline
17 & \(f_{17}(x)=\frac{1}{4000}\sum\limits_{i=1}^{D}{x_{i}^{2}-\prod\limits_{i=1}^{D }{cos(\frac{x_{i}}{\sqrt{i}})+1}}\) & Rastrigin & \([-5.12,5.12]^{D}\) \\ \hline
18 & \(f_{18}(x)=0.5+\frac{sin^{2}\sqrt{\sum\limits_{i=1}^{D}{x_{i}^{2}}}-0.5}{\left( 1+0.001\sum\limits_{i=1}^{D}{x_{i}^{2}}\right)^{2}}\) & Schaffer & \([-100,100]^{D}\) \\ \hline
19 & \(f_{19}(x)=\sum\limits_{i=1}^{D}{[100(x_{i+1}-x_{i}^{2})^{2}+\left(x_{i}-1 \right)^{2}]}\) & Rosenbrock & \([-10,10]^{D}\) \\ \hline
20 & \(f_{20}(x)=\sum\limits_{i=1}^{D}{x_{i}^{2}}\) & Sphere & \([-100,100]^{D}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: _Cont._result was recorded after separate trials for 20 times. The results are listed in Tables 3 and 4; it can be clearly seen that most results of CAABC are remarkable in accuracy of convergence. Other algorithms are run with longer CPU time, which proves the superiority of CAABC algorithm. We select five representative test function for comparison, namely, Sphere (unimodal separable function US), Rosenbrock (unimodal nonseparable function UN), Rastrigin, Alpine (multimodal separable function MS) and Ackley (multimodal nonseparable function MN). The result is given in Figures 6 and 7, which can make it visualized clearly from different views. In addition, the abscissa of the Figure 6 and 7 represents the number of iterations of the algorithm, the ordinate represents the value of the optimization function.