2.1. Proposed initialization Distribution

2.2. Chromosome rejection rule

2.3. The proposed sampling procedure

3.1. Test functions

• Bf1 (Bohachevsky 1) function:

• Bf2 (Bohachevsky 2) function:

  $$f\left(x\right)=x_1^2+2x_2^2-\frac{3}{10}cos\left(3\pi x_1\right)cos\left(4\pi x_2\right)+\frac{3}{10}$$

  with $x\in {[-50,50]}^2$.
• Branin function: $f\left(x\right)={\left(x_2-\frac{5.1}{4{\pi }^2}{x}_1^2+\frac{5}{\pi }{x}_1-6\right)}^2+10\left(1-\frac{1}{8\pi }\right)cos\left(x_1\right)+10$ with $-5\le x_1\le 10,0\le x_2\le 15$.
• CM function:

  $$f\left(x\right)=\sum _{i=1}^n{x}_i^2-\frac{1}{10}\sum _{i=1}^{n}cos\left(5\pi x_i\right)$$

  where $x\in {[-1,1]}^n$. In the conducted experiments the value $n=4$ was used.
• Camel function:

  $$f\left(x\right)=4x_1^2-2.1x_1^4+\frac{1}{3}{x}_1^6+x_1{x}_2-4x_2^2+4x_2^4,x\in {[-5,5]}^2$$
• Easom function:

  $$f\left(x\right)=-cos\left(x_1\right)cos\left(x_2\right)exp\left({\left(x_2-\pi \right)}^2-{\left(x_1-\pi \right)}^2\right)$$

  with $x\in {[-100,100]}^2.$
• Exponential function, defined as:

  $$f\left(x\right)=-exp\left(-0.5\sum _{i=1}^n{x}_i^2\right),-1\le x_i\le 1$$
  The values $n=4,8,16,32$ were used in the executed experiments.
• Griewank2 function:

  $$f\left(x\right)=1+\frac{1}{200}\sum _{i=1}^2{x}_i^2-\prod _{i=1}^2\frac{cos\left(x_i\right)}{\sqrt{\left(i\right)}},x\in {[-100,100]}^2$$
• Griewank10 function. The function is given by the equation

  $$f\left(x\right)=\sum _{i=1}^n\frac{x_i^2}{4000}-\prod _{i=1}^{n}cos\left(\frac{x_i}{\sqrt{i}}\right)+1$$

  with $n=10$.
• Gkls function. $f\left(x\right)=Gkls(x,n,w)$, is a function with w local minima, described in [72] with $x\in {[-1,1]}^n$ and n a positive integer between 2 and 100. The values $n=2,3$ and $w=50$ were used in the conducted experiments.
• Goldstein and Price function

  $$\begin{array}{ccc}\hfill f\left(x\right)& =& \left[1+{\left(x_1+x_2+1\right)}^2\right.\hfill \\ & & \left(19-14x_1+3x_1^2-14x_2+6x_1{x}_2+3x_2^2\right)]\times \hfill \\ & & [30+{\left(2x_1-3x_2\right)}^2\hfill \\ & & \left(18-32x_1+12x_1^2+48x_2-36x_1{x}_2+27x_2^2\right)]\hfill \end{array}$$
  With $x\in {[-2,2]}^2$.
• Hansen function: $f\left(x\right)={\sum }_{i=1}^{5}icos\left[(i-1)x_1+i\right]{\sum }_{j=1}^{5}jcos\left[(j+1)x_2+j\right]$, $x\in {[-10,10]}^2$ .
• Hartman 3 function:

  $$f\left(x\right)=-\sum _{i=1}^4{c}_{i}exp\left(-\sum _{j=1}^3{a}_{ij}{\left(x_j-p_{ij}\right)}^2\right)$$

  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

  $$p=\left(\begin{array}{ccc}0.3689& 0.117& 0.2673\\ 0.4699& 0.4387& 0.747\\ 0.1091& 0.8732& 0.5547\\ 0.03815& 0.5743& 0.8828\end{array}\right)$$
• Hartman 6 function:

  $$f\left(x\right)=-\sum _{i=1}^4{c}_{i}exp\left(-\sum _{j=1}^6{a}_{ij}{\left(x_j-p_{ij}\right)}^2\right)$$

  with $x\in {[0,1]}^6$ and $a=\left(\begin{array}{cccccc}10& 3& 17& 3.5& 1.7& 8\\ 0.05& 10& 17& 0.1& 8& 14\\ 3& 3.5& 1.7& 10& 17& 8\\ 17& 8& 0.05& 10& 0.1& 14\end{array}\right),c=\left(\begin{array}{c}1\\ 1.2\\ 3\\ 3.2\end{array}\right)$ and

  $$p=\left(\begin{array}{cccccc}0.1312& 0.1696& 0.5569& 0.0124& 0.8283& 0.5886\\ 0.2329& 0.4135& 0.8307& 0.3736& 0.1004& 0.9991\\ 0.2348& 0.1451& 0.3522& 0.2883& 0.3047& 0.6650\\ 0.4047& 0.8828& 0.8732& 0.5743& 0.1091& 0.0381\end{array}\right)$$
• Potential function. The molecular conformation corresponding to the global minimum of the energy of N atoms interacting via the Lennard-Jones potential[73] is used a test function here and it is defined by:

  $$V_{LJ}\left(r\right)=4ϵ\left[{\left(\frac{\sigma }{r}\right)}^{12}-{\left(\frac{\sigma }{r}\right)}^6\right]$$
  The values $N=3,5$ were used in the conducted experiments.
• Rastrigin function.

  $$f\left(x\right)=x_1^2+x_2^2-cos\left(18x_1\right)-cos\left(18x_2\right),x\in {[-1,1]}^2$$
• Rosenbrockfunction.

  $$f\left(x\right)=\sum _{i=1}^{n-1}\left(100{\left(x_{i+1}-x_i^2\right)}^2+{\left(x_i-1\right)}^2\right),-30\le x_i\le 30.$$
  The values $n=4,8,16$ were used in the conducted experiments.
• Shekel 5function.

• Shekel 7 function.

• Shekel 10 function.

• Sinusoidal function:

  $$f\left(x\right)=-\left(2.5\prod _{i=1}^{n}sin\left(x_i-z\right)+\prod _{i=1}^{n}sin\left(5\left(x_i-z\right)\right)\right),0\le x_i\le \pi .$$
  The values of $n=4,8,16$ and $z=\frac{\pi }{6}$ was used in the conducted experiments.
• Test2N function:

  $$f\left(x\right)=\frac{1}{2}\sum _{i=1}^n{x}_i^4-16x_i^2+5x_i,x_i\in [-5,5].$$
  The function has $2^n$ in the specified range and in our experiments we used $n=4,5,6,7$.
• Test30N function:

  $$f\left(x\right)=\frac{1}{10}{sin}^2\left(3\pi x_1\right)\sum _{i=2}^{n-1}\left({\left(x_i-1\right)}^2\left(1+{sin}^2\left(3\pi x_{i+1}\right)\right)\right)+{\left(x_n-1\right)}^2\left(1+{sin}^2\left(2\pi x_n\right)\right)$$

  with $x\in [-10,10]$, with ${30}^n$ local minima in the search space. For our experiments we used $n=3,4$.

3.2. Experimental results

• High Conditioned Elliptic function, defined as

  $$f\left(x\right)=\sum _{i=1}^n{\left({10}^6\right)}^{\frac{i-1}{n-1}}{x}_i^2$$
• Cm function, defined as

  $$f\left(x\right)=\sum _{i=1}^n{x}_i^2-\frac{1}{10}\sum _{i=1}^{n}cos\left(5\pi x_i\right)$$