## 5 Numerical Experiments
### Experiment Setup  
We implemented four algorithms: GA-TNE+NRO, conventional GA [1], PSO [2], and DE [3] in MATLAB and ran our experiment on a laptop with 4-core CPU in Windows 7. We applied these algorithms on a benchmark of eight nonlinear unconstrained optimization problems that are commonly used to evaluate the performance of global optimization algorithms [31]. These problems are difficult to solve using conventional optimization algorithms due to their large search space, numerous local minima, and fraudulence. Table 2 summarizes these optimization problems. For convenience, we refer to the optimization problems in Table 2 by their objective functions \(f_{1}(\mathbf{x})\) through \(f_{8}(\mathbf{x})\). Figure 5 shows the solution spaces of these problems. Problems \(f_{1}(\mathbf{x})\) through \(f_{4}(\mathbf{x})\) are uni-modal, which have precisely one global optimum, and are good for studying how well the algorithm can exploit the solution space [32]. Problems \(f_{5}(\mathbf{x})\) through \(f_{8}(\mathbf{x})\) are multimodal, which have one global optimum and many local optima. The number of local optima will increase exponentially as the number of dimensions increases. Table 3 lists the set of parameters we used to run the algorithms on the benchmark problems. Here, \(G\) denotes the number of iterations, \(N\) is the size of the initial population, \(P_{c}\) is the probability of crossover operations, \(P_{m}\) is the probability of mutation operations, and \(P_{r}\) is the probability of recombination operations. For impartial comparison, each algorithm is executed 50 times for every benchmark problem. For each run, the optimization process terminates if \(|O_{b}-O^{*}|\leq\epsilon\), where \(O_{b}\) denotes the best optimized objective function value, \(O*\) is the real optimal value, and \(\epsilon=10^{-4}\) is the precision threshold. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Function Name** & **Function Formula** & **Optimal Solution** & **Optimum** \\ \hline [PERSON] & \(\begin{matrix}\min f_{1}(\mathbf{x})=20\times(1-e^{-0.2\times\sqrt{0.5}(x_{1}^{2 }+x_{2}^{2})})\\ -e^{0.5(\cos(2\pi x1)+0.5(2\pi x2))}+\epsilon,x\in[-5,5]\end{matrix}\) & \([-3,0.5]\) & \(0\) \\ \hline [PERSON] & \(\begin{matrix}\min f_{2}(\mathbf{x})=(1.5-x_{1}+x_{1}\times x_{2})2+(2.25-x_{1} +x1\times x_{2}^{2})2\\ +(2.625-x^{1}+x^{1}\times x_{2}^{3})2,\mathbf{x}\in[-5,5]\end{matrix}\) & \([-3,0.5]\) & \(0\) \\ \hline Crossintray & \(\begin{matrix}\min f_{3}(\mathbf{x})=-0.0001\times(|\sin(x_{1})\times\sin(x_{2}) \times e^{|100-\frac{\sqrt{2}_{1}^{2}-x_{2}^{2}}{|1-1|}+1)0.1}\\ \mathbf{x}\in[-10,10]\end{matrix}\) & \([0.3459,0.3459]\) & \(0\) \\ \hline Giunta & \(\begin{matrix}\min f_{4}(\mathbf{x})=0.6+\sum\limits_{i=1}^{2}[\sin^{2}(1- \frac{16}{15}x_{i})-\frac{1}{50}\sin(4-\frac{64}{15}x_{i})\\ -\sin(1-\frac{16}{15}x_{i})],\mathbf{x}\in[-15,15]\end{matrix}\) & \([-10,10]\) & \(-20\) \\ \hline Himmelblau & \(\begin{matrix}\min f_{5}(x)=(x_{1}^{2}+x_{2}-11)^{2}+(x_{1}+x_{2}^{2}-7)^{2}, \mathbf{x}\in[-5,5]\end{matrix}\) & \([2,3]\) & \(0\) \\ \hline Penholder & \(\begin{matrix}\min f_{6}(\mathbf{x})=-e^{-1/|\cos(x_{1})\times\cos(x_{2}) \times e^{|1-\frac{\sqrt{2}_{1}^{2}+x_{2}^{2}}{2m}|}},\mathbf{x}\in[-11,11] \end{matrix}\) & \([-9.64617,9.64617]\) & \(-0.96353\) \\ \hline Testtubeholder & \(\begin{matrix}\min f_{7}(\mathbf{x})=-4|\sin(x_{1})\times\cos(x_{2})\times e^{| \cos\frac{(x_{1}^{2}-x_{2}^{2})}{2m}|},\\ x\in[-10,10]\end{matrix}\) & \([-\pi/2,0]\) & \(-10.8723\) \\ \hline Levi13 & \(\begin{matrix}\min f_{8}(\mathbf{x})=\sin^{2}(3\pi x_{1})+(x_{1}-1)2\times(1+ \sin^{2}(3\pi x_{2})+\\ (x_{2}-1)2\times(1+\sin^{2}(2\pi x_{2})),x\in[-10,10]\end{matrix}\) & \([1,1]\) & \(0\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Benchmark functions. Figure 5: Plots of benchmark functions.