# Use RBF as a Sampling Method in Multistart Global Optimization Method
## 3 Experiments
### Test Functions  
* **Bent Cigar function** The function is \[f(x)=x_{1}^{2}+10^{6}\sum_{i=2}^{n}x_{i}^{2}\] with the global minimum \(f(x^{*})=0\). For the conducted experiments the value \(n=10\) was used. * **Bf1** function. The function Bohachevsky 1 is given by the equation \[f(x)=x_{1}^{2}+2x_{2}^{2}-\frac{3}{10}\cos(3\pi x_{1})-\frac{4}{10}\cos(4\pi x _{2})+\frac{7}{10}\] with \(x\in[-100,100]^{2}\). The value of the global minimum is 0.0. * **Bf2** function. The function Bohachevsky 2 is given by the equation \[f(x)=x_{1}^{2}+2x_{2}^{2}-\frac{3}{10}\cos(3\pi x_{1})\cos(4\pi x_{2})+\frac{ 3}{10}\] with \(x\in[-50,50]^{2}\). The value of the global minimum is 0.0. * **Branin** function. The function is defined by \(f(x)=\left(x_{2}-\frac{5.1}{4\pi^{2}}x_{1}^{2}+\frac{5}{\pi}x_{1}-6\right)^{ 2}+10\Big{(}1-\frac{1}{8\pi}\Big{)}\cos(x_{1})+10\) with \(-5\leq x_{1}\leq 10\), \(0\leq x_{2}\leq 15\). The value of the global minimum is 0.397887.with \(x\in[-10,10]^{2}\). The value of the global minimum is \(-0.352386\). * **CM** function. The Cosine Mixture function is given by the equation \[f(x)=\sum_{i=1}^{n}x_{i}^{2}-\frac{1}{10}\sum_{i=1}^{n}\cos(5\pi x_{i})\] with \(x\in[-1,1]^{n}\). The value of the global minimum is \(-0.4\) and in our experiments we have used \(n=4\). The corresponding function is denoted as CM4
* **Camel** function. The function is given by \[f(x)=4x_{1}^{2}-2.1x_{1}^{4}+\frac{1}{3}x_{1}^{6}+x_{1}x_{2}-4x_{2}^{2}+4x_{2} ^{4},\quad x\in[-5,5]^{2}\] The global minimum has the value of \(f(x^{*})=-1.0316\)
* **Discus function. ** The function is defined as \[f(x)=10^{6}x_{1}^{2}+\sum_{i=2}^{n}x_{i}^{2}\] with global minimum \(f(x^{*})=0\). For the conducted experiments the value \(n=10\) was used. * **Easom** function. The function is given by the equation \[f(x)=-\cos(x_{1})\cos(x_{2})\exp\Big{(}(x_{2}-\pi)^{2}-(x_{1}-\pi)^{2}\Big{)}\] with \(x\in[-100,100]^{2}\) and global minimum \(-1.0\)* **Exponential** function. The function is given by \[f(x)=-\exp\left(-0.5\sum_{i=1}^{n}x_{i}^{2}\right),\quad-1\leq x_{i}\leq 1\] The global minimum is located at \(x^{*}=(0,0,\ldots,0)\) with value \(-1\). In our experiments we used this function with \(n=4,16,64\) and the corresponding functions are denoted by the labels EXP4, EXP16, EXP64. * **Griewank2** function. The function is given by \[f(x)=1+\frac{1}{200}\sum_{i=1}^{2}x_{i}^{2}-\prod_{i=1}^{2}\frac{\cos(x_{i})}{ \sqrt{(i)}},\quad x\in[-100,100]^{2}\] The global minimum is located at the \(x^{*}=(0,0,\ldots,0)\) with value 0. * **Griewank10** function. The function is given by the equation \[f(x)=\sum_{i=1}^{n}\frac{x_{i}^{2}}{4000}-\prod_{i=1}^{n}\cos\left(\frac{x_{i} }{\sqrt{i}}\right)+1\] In our experiments we have used \(n=10\) and the global minimum is 0.0. The function has several local minima in the specified range. * **[PERSON]** function. \(f(x)=\sum_{i=1}^{5}i\cos[(i-1)x_{1}+i]\sum_{j=1}^{5}j\cos[(j+1)x_{2}+j]\), \(x\in[-10,10]^{2}\). The global minimum of the function is \(-176.541793\). * **[PERSON] 3** function. The function is given by \[f(x)=x_{1}^{2}+2x_{2}^{2}-\frac{3}{10}\cos(3\pi x_{1})-\frac{4}{10}\cos(4\pi x _{2})+\frac{7}{10}\]0 with \(x\in[0,1]^{3}\) and \(a=\left( \begin{array}{ccc}3&10&30\\ 0.1&10&35\\ 3&10&30\\ 0.1&10&35\end{array} \right)\), \(c=\left( \begin{array}{c}1\\ 1.2\\ 3\\ 3.2\end{array} \right)\) and \[p=\left( \begin{array}{ccc}0.3689&0.117&0.2673\\ 0.4699&0.4387&0.747\\ 0.1091&0.8732&0.5547\\ 0.03815&0.5743&0.8828\end{array} \right)\] The value of the global minimum is \(-3.862782\). * **Hartman 6** function. \[f(x)=-\sum_{i=1}^{4}c_{i}\exp\left(-\sum_{j=1}^{6}a_{ij}\big{(}x_{j}-p_{ij} \big{)}^{2}\right)\] with \(x\in[0,1]^{6}\) and \(a=\left( \begin{array}{ccccc}1&3&17&3.5&1.7&8\\ 0.05&10&17&0.1&8&14\\ 3&3.5&1.7&10&17&8\\ 17&8&0.05&10&0.1&14\end{array} \right)\), \(c=\left( \begin{array}{c}1\\ 1.2\\ 3\\ 3.2\end{array} \right)\) and \[p=\left(