# Simulated Annealing with Exploratory Sensing for Global Optimization
## Appendix A Classical Test Functions
### Schwefel Function (\(f_{2}\))
: \(f_{18}(\mathbf{x})=(1+(x_{1}+x_{2}+1)^{2}(19-14x_{1}+13x_{1}^{2}-14x_{2}+6x_{1}x_ {2}+3x_{2}^{2}))\)__  
\(*\left(30+(2x_{1}-3x_{2})^{2}(18-32x_{1}+12x_{1}^{2}-48x_{2}-36x_{1}x_{2}+27x_{2 }^{2})\right)\)_._  
**Search space:**__\(-2\leq x_{i}\leq 2,\ i=1,2\)_._  
**Global minimum:**__\(\mathbf{x}^{*}=(0,-1)\)_;_ \(f_{18}(\mathbf{x}^{*})=3\)_._  
**Remark A.18**. : _Hartmann Function (\(f_{19}\))_  
**Definition A18**. : \(f_{19}(\mathbf{x})=-\sum_{i=1}^{4}\alpha_{i}\exp\left[-\sum_{j=1}^{3}A_{ij} \left(x_{j}-P_{ij}\right)^{2}\right],\)__  
\(\alpha=[1,1.2,3,3.2]^{T}\)_,_ \(A=\left[ \begin{array}{cccc}3.0&10&30\\ 0.1&10&35\\ 3.0&10&30\\ 0.1&10&35\end{array} \right]\)_,_ \(P=10^{-4}\left[ \begin{array}{cccc}6890&1170&2673\\ 4699&4387&7470\\ 1091&8732&5547\\ 381&5743&8828\end{array} \right].\)__  
**Search space:**__\(0\leq x_{i}\leq 1,\ i=1,2,3\)_._  
**Global minimum:**__\(\mathbf{x}^{*}=(0.114614,0.555649,0.852547);f_{19}(\mathbf{x}^{*})=-3.86278\)_._  
**Remark A.19**. : _Hartmann Function (\(f_{20}\))_  
**Definition A19**. : \(f_{20}(\mathbf{x})=-\sum_{i=1}^{4}\alpha_{i}\exp\left[-\sum_{j=1}^{6}B_{ij} \left(x_{j}-Q_{ij}\right)^{2}\right],\)__  
\(\alpha=[1,1.2,3,3.2]^{T}\)_,_ \(B=\left[ \begin{array}{cccc}10&3&17&3.05&1.7&8\\ 0.05&10&17&0.1&8&14\\ 3&3.5&1.7&10&17&8\\ 17&8&0.05&10&0.1&14\end{array} \right]\)_,_  
\(Q=10^{-4}\left[ \begin{array}{cccc}1312&1696&5569&124&8283&5886\\ 2329&4135&8307&3736&1004&9991\\ 2348&1451&3522&2883&3047&6650\\ 4047&8828&8732&5743&1091&381\end{array} \right]\)_._  
**Search space:**__\(0\leq x_{i}\leq 1,\ i=1,\ldots,6\)_._  
**Global minimum:**__\(\mathbf{x}^{*}=(0.201690,0.150011,0.476874,0.275332,0.311652,0.657300);\)__  
\(f_{20}(\mathbf{x}^{*})=-3.32237\)_._  
**Remark A.20**. : _Shekel Functions (\(f_{21},f_{22},f_{23}\))_  
**Definition A20**. : \(f_{21}(\mathbf{x})=S_{4,5}(\mathbf{x}),f_{22}(\mathbf{x})=S_{4,7}(\mathbf{x}),f_{23}(\mathbf{x})=S_{4,10}(\mathbf{x}),\)__  
_where \(\mathrm{S}_{4,m}(\mathbf{x})=-\sum_{j=1}^{m}\left[\sum_{i=1}^{4}\left(x_{i}-C_{ ij}\right)^{2}+\beta_{j}\right]^{-1}\), \(m=5,7,10\),_  
\(\beta=\frac{1}{10}\left[1,2,2,4,4,6,3,7,5,5\right]^{T}\)_,_ \(C=\left[ \begin{array}{cccccccc}4.0&1.0&8.0&6.0&3.0&2.0&5.0&8.0&6.0&7.0\\ 4.0&1.0&8.0&6.0&7.0&9.0&5.0&1.0&2.0&3.6\\ 4.0&1.0&8.0&6.0&3.0&2.0&3.0&8.0&6.0&7.0\\ 4.0&1.0&8.0&6.0&7.0&9.0&3.0&1.0&2.0&3.6\end{array} \right]\)_._  
**Search space:**__\(0\leq x_{i}\leq 10,\ i=1,\ldots,4\)_._  
**Global minimum:**__\(\mathbf{x}^{*}=(4,4,4,4)\)_;_  
\(f_{21}(\mathbf{x}^{*})=-10.1532\)_,_  
\(f_{22}(\mathbf{x}^{*})=-10.4029\)_,_  
\(f_{23}(\mathbf{x}^{*})=-10.5364\)_._  
**Remark A.21**. : _Function (\(f_{24}\))_  
**Definition A21**. : \(f_{24}(\mathbf{x})=-\sum_{i=1}^{n}\sin(x_{i})\sin^{20}(\frac{ix_{i}^{2}}{n})\)._  
**Search space: \(0\leq x_{i}\leq\pi,\ i=1,\ldots,n\). **  
**Global minimum: \(f_{24}(\mathbf{x}^{*})=-99.2784\). **  
**Appendix A.22**: _Function (\(f_{25}\))_  
**Definition A22**: \(f_{25}(\mathbf{x})=\frac{1}{n}\sum_{i=1}^{n}(x_{i}^{4}-16x_{i}^{2}+5x_{i})\). **  
**Search space: \(-5\leq x_{i}\leq 5,\ i=1,\ldots,n\). **  
**Global minimum: \(f_{25}(\mathbf{x}^{*})=-78.33236\). **