# Advanced Parameter-Setting-Free Harmony Search Algorithm
## 4 Performance Comparison by PSF Type
### Test in Discrete Test Function  
We compared the advanced PSF HS results with those using the test function from the conventional PSF scheme. \[f(x)=\sum_{i=1}^{5}(x_{i}-i)^{2} \tag{13}\]  
The input range of Equation (13) is \(x_{i}\in\{1,\ 2,\ \dots,\ 10\}\) as discrete values, and the minimum value is \(f(x^{*})=0,\ x^{*}=(1,\ 2,\ 3,\ 4,\ 5)\). When the HS algorithm is performed using this equation, the HMS is equal to 25 and \(\pm 1\) is set if there is a change due to the PAR. Unlike the conventional PSF scheme, one of the advanced PSF methods can be used only when the number of executions is determined. Therefore, the maximum number of iterations was set to 500 and 1000, and when the minimum value of 0 was obtained, a total of 100,000 results were collected, and the number of iterations necessary for the minimum value was measured. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Function** & **PSF Type** & **4000 iter** & **8000 iter** & **12,000 iter** & **16,000 iter** & **20,000 iter** \\ \hline Sphere & Conventional & 7.3047 & 1.7931 & 1.1941 & 0.9677 & 0.8475 \\  & Iteration & **2.5973** & **0.7254** & **0.1530** & **0.0105** & **2.1888 \(\mathbf{\times\ 10^{4}}\)** \\  & Object & 9.0669 & 6.1348 & 3.8778 & 2.2569 & 1.1523 \\  & Object + \(\mathbf{b}\) & 3.5976 & 2.1292 & 1.3533 & 0.9450 & 0.7000 \\ Rosenbrock & Conventional & \(4.2053\times 10^{5}\) & \(1.9097\times 10^{3}\) & \(8.6170\times 10^{2}\) & \(6.2643\times 10^{2}\) & \(5.0953\times 10^{2}\) \\  & Iteration & **2.4875 \(\mathbf{\times\ 10^{3}}\)** & \(6.1813\times\mathbf{10^{2}}\) & \(2.7474\times\mathbf{10^{2}}\) & \(1.8004\times\mathbf{10^{2}}\) & \(1.3778\times\mathbf{10^{2}}\)** \\  & Object & \(1.8541\times 10^{4}\) & \(1.1280\times 10^{4}\) & \(6.6921\times 10^{3}\) & \(3.8371\times 10^{3}\) & \(2.1353\times 10^{3}\) \\  & Object + \(\mathbf{b}\) & \(1.1240\times 10^{4}\) & \(9.0852\times 10^{3}\) & \(7.5634\times 10^{3}\) & \(6.4223\times 10^{3}\) & \(5.5621\times 10^{3}\) \\ Rastringin & Conventional & 46.1926 & 30.3153 & 26.3731 & 24.1580 & 22.5746 \\  & Iteration & 25.7417 & **13.8458** & **11.2223** & **8.9907** & **7.5238** \\  & Object & 42.7380 & 35.5440 & 34.2143 & 33.1040 & 31.9219 \\  & Object + \(\mathbf{b}\) & **25.5767** & 24.9163 & 24.3740 & 23.9071 & 23.4128 \\ Ackley & Conventional & 9.7717 & 6.2214 & 5.2261 & 4.7196 & 4.3750 \\  & Iteration & **7.0990** & **5.0225** & 4.0665 & 3.4594 & 3.0939 \\  & Object & 16.7124 & 7.5771 & 5.0366 & 4.6922 & 4.4424 \\  & Object + \(\mathbf{b}\) & 16.7367 & 6.9600 & **1.4705** & **1.2630** & **1.2277** \\ Happycat & Conventional & 0.5862 & 0.5011 & 0.4521 & 0.4153 & 0.3820 \\  & Iteration & 0.3783 & 0.2216 & 0.1605 & **0.1282** & **0.1076** \\  & Object & **0.3581** & **0.2064** & **0.1548** & 0.1303 & 0.1164 \\  & Object + \(\mathbf{b}\) & 0.3715 & 0.2116 & 0.1553 & 0.1295 & 0.1144 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Average of 10,000 results for 32 variables in each function using PSF HS. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Function Name** & **Function** & **Input Range** & **Minimum** \\ \hline Sphere & \(f(x)=\sum\limits_{i=1}^{n}x_{i}^{2}\) & \([-5.12,5.12]\) & 0 \\ [PERSON] & \(f(x)=\sum\limits_{i=1}^{n}\left[100\left(x_{i+1}-x_{i}^{2}\right)^{2}+(1-x_{i})^{ 2}\right]\) & \([-5,10]\) & 0 \\ [PERSON] & \(f(x)=10n+\sum\limits_{i=1}^{n}\left(x_{i}^{2}-10\cos(2\pi x_{i})\right)\) & \([-5.12,5.12]\) & 0 \\ Ackley & \(f(x)=20+e-20^{-2\sqrt{\frac{1}{2}\sum\limits_{i=1}^{n}x_{i}^{2}}}-e^{\frac{1}{2} \sum\limits_{i=1}^{n}\cos(2\pi x_{i})}\) & \([-32,32]\) & 0 \\ Happycat & \(f(x)=\left(\|\mathbf{x}\|^{2}-n\right)^{\frac{1}{2}}+\frac{1}{n}\left(\frac{1}{2} \|\mathbf{x}\|^{2}+\sum\limits_{i=1}^{n}x_{i}\right)+\frac{1}{2}\) & \([-2,2]\) & 0 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Test functions. Figure 12 shows the results of Equation (13) using conventional PSF HS. The initial number of iterations required for the existing PSF configuration was specified as 50.