# Use RBF as a Sampling Method in Multistart Global Optimization Method
## 3 Experiments
### Test Functions
\begin{array}{ccccc}0.1312&0.1696&0.5569&0.0124&0.8283&0.5886\\ 0.2329&0.4135&0.8307&0.3736&0.1004&0.9991\\ 0.2348&0.1451&0.3522&0.2883&0.3047&0.6650\\ 0.4047&0.8828&0.8732&0.5743&0.1091&0.0381\end{array} \right)\]the value of the global minimum is \(-3.322368\). * **High Conditioned Elliptic** function, defined as \[f(x)=\sum_{i=1}^{n}\Bigl{(}10^{6}\Bigr{)}^{\frac{i-1}{n-1}}x_{i}^{2}\] with global minimum \(f(x^{*})=0\) and the value \(n=10\) was used in the conducted experiments
* **Potential** function. The molecular conformation corresponding to the global minimum of the energy of N atoms interacting via the Lennard-Jones potential [69] is used as a test case here. The function to be minimized is given by: \[V_{LJ}(r)=4\epsilon\biggl{[}\Bigl{(}\frac{\sigma}{r}\Bigr{)}^{12}-\Bigl{(} \frac{\sigma}{r}\Bigr{)}^{6}\biggr{]}\] (9) In the current experiments two different cases were studied: \(n=3,\ 5\)
* **Rastrigin** function. The function is given by \[f(x)=x_{1}^{2}+x_{2}^{2}-\cos(18x_{1})-\cos(18x_{2}),\quad x\in[-1,1]^{2}\] The global minimum is located at \(x^{*}=(0,0)\) with value \(-2.0\). * **Shekel 7** function. \[f(x)=-\sum_{i=1}^{7}\frac{1}{(x-a_{i})(x-a_{i})^{T}+c_{i}}\] with \(x\in[0,10]^{4}\) and \(a=\left( \begin{array}{cccc}4&4&4&4\\ 1&1&1&1\\ 8&8&8&8\\ 6&6&6&6\\ 3&7&3&7\\ 2&9&2&9\\ 5&3&5&3\end{array} \right)\), \(c=\left( \begin{array}{c}0.1\\ 0.2\\ 0.2\\ 0.4\\ 0.6\\ 0.3\end{array} \right)\). The value of the global minimum is \(-10.342378\). * **Shekel 5** function. \[f(x)=-\sum_{i=1}^{5}\frac{1}{(x-a_{i})(x-a_{i})^{T}+c_{i}}\] with \(x\in[0,10]^{4}\) and \(a=\left( \begin{array}{cccc}4&4&4&4\\ 1&1&1&1\\ 8&8&8&8\\ 6&6&6&6\\ 3&7&3&7\end{array} \right)\), \(c=\left( \begin{array}{c}0.1\\ 0.2\\ 0.2\\ 0.4\\ 0.4\end{array} \right)\). The value of the global minimum is \(-10.107749\). * **Shekel 10** function. \[f(x)=-\sum_{i=1}^{10}\frac{1}{(x-a_{i})(x-a_{i})^{T}+c_{i}}\]\[\text{with }x\in[0,10]^{4}\text{ and }a=\left( \begin{array}{cccc}4&4&4&4\\ 1&1&1&1\\ 8&8&8&8\\ 6&6&6&6\\ 3&7&3&7\\ 2&9&2&9\\ 5&5&3&3\\ 8&1&8&1\\ 6&2&6&2\\ 7&3.6&7&3.6\end{array} \right),\ c=\left( \begin{array}{c}0.1\\ 0.2\\ 0.4\\ 0.4\\ 0.4\\ 0.6\\ 0.3\\ 0.7\\ 0.5\\ 0.6\end{array} \right).\text{ The value of the global minimum is }-10.536410. * **Sinusoidal** function. The function is given by \[f(x)=-\left(2.5\prod_{i=1}^{n}\sin(x_{i}-z)+\prod_{i=1}^{n}\sin(5(x_{i}-z)) \right),\quad 0\leq x_{i}\leq\pi.\] The global minimum is located at \(x^{*}=(2.09435,2.09435,\ldots,2.09435)\) with \(f(x^{*})=-3.5\). In our experiments we used \(n=4,8,16\) and \(z=\frac{\pi}{6}\) and the corresponding functions are denoted by the labels SINU4, SINU8 and SINU16 respectively. * **Test2N** function. This function is given by the equation \[f(x)=\frac{1}{2}\sum_{i=1}^{n}x_{i}^{4}-16x_{i}^{2}+5x_{i},\quad x_{i}\in[-5,5].\] The function has \(2^{n}\) in the specified range and in our experiments we used \(n=4,5,6,7\). The corresponding values of the global minimum is \(-156.664663\) for \(n=4\), \(-195.830829\) for \(n=5\), \(-234.996994\) for \(n=6\) and \(-274.163160\) for \(n=7\). * **Test30N** function. This function is given by \[f(x)=\frac{1}{10}\sin^{2}(3\pi x_{1})\sum_{i=2}^{n-1}\Bigl{(}(x_{i}-1)^{2} \Bigl{(}1+\sin^{2}(3\pi x_{i+1})\Bigr{)}\Bigr{)}+(x_{n}-1)^{2}\Bigl{(}1+\sin^{ 2}(2\pi x_{n})\Bigr{)}\] with \(x\in[-10,10]\). The function has \(30^{n}\) local minima in the specified range and we used \(n=3,4\) in our experiments. The value of the global minimum for this function is \(0.0\).