# Simulated Annealing with Exploratory Sensing for Global Optimization
## Appendix A Classical Test Functions
### Schwefel Function (\(f_{2}\))  
**Definition A2**. : \(f_{2}(\mathbf{x})=\sum_{i=1}^{n}|x_{i}|+\Pi_{i=1}^{n}|x_{i}|\)_._  
**Search space:**\(-10\leq x_{i}\leq 10\)_,_ \(i=1,\ldots,n\)  
**Global minimum:**\(\mathbf{x}^{*}=(0,\ldots,0)\)_,_ \(f_{2}(\mathbf{x}^{*})=0\)_._ \begin{table}
\begin{tabular}{l l l l l l l} \hline \hline
**Criterion** & **Compared Methods** & **No. of Beats** & \(R^{+}\) & \(R^{-}\) & \(p\)**-Value** & **Best Method** \\ \hline Average Errors & BLX-GLS50 & SAES & 16/8 & 133.5 & 191.5 & 0.9458 & – \\  & BLX-MA & SAES & 11/14 & 209 & 116 & 0.7415 & – \\  & CoEVO & SAES & 10/15 & 232 & 93 & 0.6344 & – \\  & DE & SAES & 17/7 & 88.5 & 236.5 & 0.5935 & – \\  & DMS-L-PSO & SAES & 13/11 & 151.5 & 173.5 & 0.5802 & – \\  & EDA & SAES & 12/12 & 169.5 & 155.5 & 1.0000 & – \\  & IPOP-CMA-ES & SAES & 18/4 & 59 & 266 & 0.3361 & – \\  & K-PCX & SAES & 11/13 & 205.5 & 119.5 & 0.8537 & – \\  & LR-CMA-ES & SAES & 13/11 & 176.5 & 148.5 & 0.7634 & – \\  & L-SADE & SAES & 16/7 & 113.5 & 211.5 & 0.5605 & – \\  & SPC-PNNX & SAES & 15/9 & 133 & 192 & 0.9304 & – \\  & cHS & SAES & 3/22 & 311 & 14 & 0.0055 & SAES \\  & HC & SAES & 2/23 & 318 & 7 & 0.0034 & SAES \\  & \(\beta\)HC & SAES & 11/13 & 205.5 & 119.5 & 0.4788 & – \\ \hline Function Evaluations & All methods & SAES & 0/25 & 325 & 0 & 2.77E–12 & SAES \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Wilcoxon rank-sum test for the compared results in Table 7.  
_Appendix A.3. Schwefel Function (\(f_{3}\))_  
**Definition A3**. : \(f_{3}(\mathbf{x})=\sum_{i=1}^{n}(\sum_{j=1}^{i}x_{j})^{2}\)_._  
**Search space: \(-100\leq x_{i}\leq 100,\ i=1,\ldots,n\) Global minimum: \(\mathbf{x}^{*}=(0,\ldots,0)\), \(f_{3}(\mathbf{x}^{*})=0\)._  
_Appendix A.4. Schwefel Function (\(f_{4}\))_  
**Definition A4**. : \(f_{4}(\mathbf{x})=\max_{i=1,\ldots n}\{|x_{i}|\}\)_._  
**Search space: \(-100\leq x_{i}\leq 100,\ i=1,\ldots,n\) Global minimum: \(\mathbf{x}^{*}=(0,\ldots,0)\), \(f_{4}(\mathbf{x}^{*})=0\)._  
_Appendix A.5. Rosenbrock Function (\(f_{5}\))_  
**Definition A5**. : \(f_{5}(\mathbf{x})=\sum_{i=1}^{n-1}\left[100\left(x_{i}^{2}-x_{i+1}\right)^{2} +\left(x_{i}-1\right)^{2}\right].\)__  
**Search space: \(-30\leq x_{i}\leq 30,\ i=1,2,\ldots,n\) Global minimum: \(\mathbf{x}^{*}=(1,\ldots,1)\), \(f_{5}(\mathbf{x}^{*})=0\)._  
_Appendix A.6. Step Function (\(f_{6}\))_  
**Definition A6**. : \(f_{6}(\mathbf{x})=\sum_{i=1}^{n}(\lfloor x_{i}+0.5\rfloor)^{2}\)_._  
**Search space: \(-100\leq x_{i}\leq 100,\ i=1,2,\ldots,n\) Global minimum: \(\mathbf{x}^{*}=(0,\ldots,0)\), \(f_{6}(\mathbf{x}^{*})=0\)._  
_Appendix A.7. Quadratic Function with Noise (\(f_{7}\))_  
**Definition A7**. : \(f_{7}(\mathbf{x})=\sum_{i=1}^{n}ix_{i}^{4}+random[0,1)\)_._  
**Search space: \(-1.28\leq x_{i}\leq 1.28,\ i=1,\ldots,n\) Global minimum: \(\mathbf{x}^{*}=(0,\ldots,0)\), \(f_{7}(\mathbf{x}^{*})=0\)._  
_Appendix A.8. Schwefel Functions (\(f_{8}\))_  
**Definition A8**. : \(f_{8}(\mathbf{x})=-\sum_{i=1}^{n}\left(x_{i}\sin\sqrt{|x_{i}|}\right).\)__  
**Search space: \(-500\leq x_{i}\leq 500,\ i=1,2,\ldots,n\). Global minimum: \(\mathbf{x}^{*}=(420.9687,\ldots,420.9687)\), \(f_{8}(\mathbf{x}^{*})=-418.9829n\)._  
_Appendix A.9. Rastrigin Function (\(f_{9}\))_  
**Definition A9**. : \(f_{9}(\mathbf{x})=10n+\sum_{i=1}^{n}\left(x_{i}^{2}-10\cos\left(2\pi x_{i} \right)\right).\)__  
**Search space: \(-5.12\leq x_{i}\leq 5.12,\ i=1,\ldots,n\). Global minimum: \(\mathbf{x}^{*}=(0,\ldots,0)\), \(f_{9}(\mathbf{x}^{*})=0\)._  
_Appendix A.10. Ackley Function (\(f_{10}\))_  
**Definition A10**. : \(f_{10}(\mathbf{x})=20+e-20e^{-\frac{1}{8}\sqrt{\frac{1}{8}\sum_{i=1}^{n}x_{i}^ {2}}}-e^{\frac{1}{8}\sum_{i=1}^{n}\cos(2\pi x_{i})}.\)__  
**Search space: \(-32\leq x_{i}\leq 32,\ i=1,2,\ldots,n\). Global minimum: \(\mathbf{x}^{*}=(0,\ldots,0)\); \(f_{10}(\mathbf{x}^{*})=0\)._  
**Remark A.17**. : _Goldstein & Price Function (\(f_{18}\))_  
**Definition A17**.