# Kernel Clustering with a Differential Harmony Search Algorithm for Scheme Classification
## 4 Experiments  
### Numerical Experiments of DHS  
#### 4.1.1 Benchmark Function Tests  
To evaluate the performance of the proposed DHS, we chose several famous typical benchmark functions, shown in Table 1. These functions are tested by simulating the HS, IHS, GHS, CHS, and the proposed DHS algorithms. Of the functions in Table 1, the Ackley function, Griewank function, Rastrigin function, and Rosenbrock function are multimodal functions. The Sphere function and Schwefel 2.22 function are unimodal functions. Parameters of the algorithms are shown in Table 2. All of the experiments are performed using a computer with 3.80 GHz AMD Athlon x4 760k with 8 GB RAM. The source code is compiled with Java SE8. We evaluated the optimization algorithms based on 10 dimensional versions of the benchmark functions. The maximum evaluation count (NI) is set to 100,000. Each function is tested 100 times for each algorithm. Table 3 shows the maximum, minimum, means, and standard deviation of errors of the algorithms on each benchmark function. Table 4 shows the distribution of the results simulated by the algorithms. As demonstrated in Tables 3 and 4, it is clear that DHS outperforms the other variants of HS in almost all of the functions except the Rosenbrock function. In these cases, the minimum, maximum, means, and standard deviations obtained by DHS are smaller than the results obtained by HS and its variants. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Function** & **Formula** & **Search Domain** & **Optimum** \\ \hline Ackley function & \(f(x_{1},\cdots,x_{n})=-20\exp\left(-0.2\sqrt{\frac{1}{n}\sum\limits_{i=1}^{n}x _{i}^{2}}\right)\) & \(-40\leq x_{i}\leq 40\) & \(f(0,\cdots,0)=0\) \\  & \(-\exp\left(\frac{1}{n}\sum\limits_{i=1}^{n}\cos(2\pi x_{i})\right)+e+20\) & & \\ \hline Griewank function & \(f(x_{1},\cdots,x_{n})=1+\frac{1}{4000}\sum\limits_{i=1}^{n}x_{i}^{2}\) & & \\  & \(-\prod\limits_{i=1}^{n}\cos\left(\frac{x_{i}}{\sqrt{i}}\right)\) & \(-600\leq x_{i}\leq 600\) & \(f(0,\cdots,0)=0\) \\ \hline Rastrigin function & \(f(x_{1},\cdots,x_{n})=An\) & \(A=10\) & \(f(0,\cdots,0)=0\) \\  & \(+\sum\limits_{i=1}^{n}\left[x_{i}^{2}-A\cos(2\pi x_{i})\right]\) & \(-5.12\leq x_{i}\leq 5.12\) & \(f(0,\cdots,0)=0\) \\ \hline Rosenbrock function & \(f(x_{1},\cdots,x_{n})=\) & \(-\infty\leq x_{i}\leq\infty\) & \(f(1,\cdots,1)=0\) \\ \hline Sphere function & \(f(x_{1},\cdots,x_{n})=\sum\limits_{i=1}^{n}x_{i}^{2}\) & \(-5.12<x_{i}<5.12\) & \(f(0,\cdots,0)=0\) \\ \hline Schwefel 2.22 function & \(f(x_{1},\cdots,x_{n})=\sum\limits_{i=1}^{n}\left|x_{i}\right|+\prod\limits_{i= 0}^{n}\left|x_{i}\right|\) & \(-10<x_{i}<10\) & \(f(0,\cdots,0)=0\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Test functions. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Algorithm** & _hms_ & _hmcr_ & _par_ & _fw_ & **NI** \\ \hline HS & 50 & 0.9 & 0.3 & 0.005 & 100,000 \\ IHS & 50 & 0.9 & Max: 0.5 Min: 0.1 & Max: 0.01 Min: 0.001 & 100,000 \\ GHS & 50 & 0.9 & 0.3 & 0.005 & 100,000 \\ CHS & 50 & 0.9 & 0.3 & 0.005 & 100,000 \\ DHS & 50 & 0.9 & 0.3 & 0.005 & 100,000 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Parameters of the algorithms. When dealing with the Rosenbrock function, the global minimum of the Rosenbrock function is inside a long, narrow, parabolic-shaped flat valley, meaning that convergence to the global minimum is difficult. Although the minimum of GHS is smaller than DHS, the maximum, mean, and standard deviation show that DHS is more stable than GHS.