# Game Theory-Inspired Evolutionary Algorithm for Global Optimization
## 4 Performance Comparison and Experimental Results
### Results and Comparision Analysis  
For each test problem, 50 independent experiments have been executed and the averaged results are shown in Table 3.  
For functions \(f_{1}\), \(f_{2}\), and \(f_{4}\), the GameEA obtains the best results with an average result of \(4.33\times 10^{-96}\), \(1.52\times 10^{-66}\), and \(0.0374\), respectively. The DPGA shows the second-best results with an average result of \(1.47\times 10^{-52}\) for function \(f_{1}\), and the RTS is third on the list. The StGA convergence accuracy is slightly worse than that of the IMGA for functions \(f_{1}\) and \(f_{2}\), but the IMGA is worse than the StGA for function \(f_{4}\). \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**No.** & **Name** & \(n\) & **Function** & **Range** \\ \hline \(f_{1}\) & Sphere & 30 & \(f(x)=\sum_{i=1}^{n}x_{i}^{2}\) & \(x_{i}\in[-100,100]\) \\ \(f_{2}\) & Schwefel 2.22 & 30 & \(f(x)=\sum_{i=1}^{n}|x_{i}^{-1}+\prod_{i=1}^{n}|x_{i}|\) & \(x_{i}\in[-10,10]\) \\ \(f_{3}\) & Schwefel 2.21 & 30 & \(f(x)=\sum_{i=1}^{n}|x_{i}|\leq i\leq n\) & \(x_{i}\in[-100,100]\) \\ \(f_{4}\) & Rosenbrock & 30 & \(f(x)=\sum_{i=1}^{n-1}|100(x_{i+1}-x_{i}^{2})^{2}+(1-x_{i})^{2}|\) & \(x_{i}\in[-30,30]\) \\ \(f_{5}\) & Step & 30 & \(f(x)=\sum_{i=1}^{n}(|x_{i}+0.5|^{2})\) & \(x_{i}\in[-100,100]\) \\ \(f_{6}\) & Noisy Quartic & 30 & \(f(x)=\sum_{i=1}^{n}x_{i}^{2}+\text{random}(0,1)\) & \(x_{i}\in[-1.28,1.28]\) \\ \(f_{7}\) & Goldstein-price & 2 & \(f(x)=(1+x_{1}+x_{2}+1)^{2}(19-14x_{1}+31x_{1}^{2}-14x_{2}+6x_{1}x_{2}+3x_{2}^{ 2}))g(x)\) & \(x_{i}\in[-2,2]\) \\  & & \(g(x)=30+(2x_{1}-3x_{2})^{2}(18-3x_{1}+12x_{1}^{2}+48x_{2}-36x_{1}x_{2}+2x_{2}^ {2})\) & \(x_{i}\in[-5,10]\) \\ \(f_{8}\) & Branin & 2 & \(f(x)=\left(x_{2}-\frac{5.1x_{1}}{4x^{2}}+\frac{5.7x_{1}}{5x_{1}}-6\right)+10 \left(1-\frac{1}{8n}\right)\cos x_{1}+10\) & \(x_{2}\in[0,15]\) \\ \(f_{9}\) & Six-hump & 2 & \(f(x)=4x_{1}^{2}-2.1x_{1}^{4}+x_{1}x_{2}-4x_{2}^{2}+4x_{2}^{4}\) & \(x_{i}\in[-5,5]^{n}\) \\ \(f_{10}\) & Rastrigin & 30 & \(f(x)=10n+\frac{x_{1}^{n}}{4(x_{1}^{2})}-10\cos(2\pi x_{i})\) & \(x_{i}\in[-5.12,5.12]\) \\ \(f_{11}\) & Griewank & 30 & \(f(x)=1+\frac{x_{1}^{n}(x_{1}^{2})}{4(x_{1}^{2})}-\frac{1}{n}\cos\left(\frac{x_{ i}}{\sqrt{t}}\right)\) & \(x_{i}\in[-600,600]\) \\ \(f_{12}\) & Schwefel 2.26 & 30 & \(f(x)=-\sum_{i=1}^{n}\left(x_{i}\sin\sqrt{|x_{i}|}\right)\) & \(x_{i}\in[-500,500]\) \\ \(f_{13}\) & [PERSON] & 30 & \(f(x)=20+\text{e}-20\exp\left(-\frac{1}{5}\sqrt{\frac{1}{n}\sum_{i=1}^{n}\left(x _{i}^{2}\right)}\right)-\exp\left(\frac{1}{n}\sum_{i=1}^{n}\cos(2\pi x_{i})\right)\) & \(x_{i}\in[-32,32]\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Test functions used for the experiments. For function \(f_{4}\), the RTS with an average of 1.391 is the silver medalist, but the GameEA's average is less than 3% of the RTS. For function \(f_{6}\), the GameEA is significantly better than the StGA and the DPGA. The GameEA is similar to the IMGA and the RTS, but it has a smaller standard deviation. For function \(f_{5}\) and \(f_{7}\), all the mentioned algorithms above converge to the global minimum. To make a careful observation of the results on the highly-multimodal functions \(f_{8}\)-\(f_{13}\), which have a large number of local optima, all the algorithms converge to the global minimum on the functions \(f_{8}\) and \(f_{9}\). The RTS, the DPGA and the GameEA show no significant difference with the convergence of the global optimum in each independent trial on function \(f_{10}\), which means that all the algorithms can solve such problems. The GameEA won a landslide victory with the best stability convergence in 50 independent trials for the functions \(f_{11}\)-\(f_{13}\), which demonstrates the search ability, stability, and robustness.