# A Novel Optimization Algorithm Combing
## 4 Experiments
### Performance Comparison between ABC, GABC, and GABCG  
Comparing the proposed algorithm with the other two algorithms on 10 benchmark functions, Table 3 shows the mean and standard deviations of each benchmark function results. The average convergence curves of representative cases are presented in Figure 3. It can be seen that the program using the GABCG algorithm gets better results on all 10 functions than the ABC algorithm and the GABC algorithm. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \(f\) & **Function** & **C** & **Initial Range** & \(f(x^{4})\) \\ \hline
**Rosenbrock** & \(f_{1}(x)=\sum\limits_{i=1}^{D-1}\left[100\left(x_{i+1}-x_{i}^{2}\right)^{2}+(x_ {i}-1)^{2}\right]\) & UN & [–50,50] & \(f(\overrightarrow{1})=0\) \\
**Sphere** & \(f_{2}(x)=\sum\limits_{i=1}^{D}x_{i}^{2}\) & US & [–100,100] & \(f(\overrightarrow{0})=0\) \\
**Elliptic** & \(f_{3}(x)=\sum\limits_{i=1}^{D}\left(10^{6}\right)^{(i-1)/(D-1)}x_{i}^{2}\) & UN & [–100,100] & \(f(\overrightarrow{0})=0\) \\
**SumSquares** & \(f_{4}(x)=\sum\limits_{i=1}^{D}ix_{i}^{2}\) & US & [–10,10] & \(f(\overrightarrow{0})=0\) \\
**Quartic** & \(f_{5}(x)=\sum\limits_{i=1}^{D}ix_{i}^{4}\) & US & [–1.28,1.28] & \(f(\overrightarrow{0})=0\) \\
**Himmelblau** & \(f_{6}(x)=\sum\limits_{i=1}^{D}\left(x_{i}^{4}-16x_{i}^{2}+5x_{i}\right)\) & MS & [–5,5] & \(f(\overrightarrow{2.9035})=78.3323\) \\
**Schaffer** & \(f_{7}(x)=0.5+\frac{\sin^{2}\left(\sqrt{\sum\limits_{i=1}^{D}x_{i}^{2}}\right) -0.5}{\left(1+0001\left(\sum\limits_{i=1}^{D}x_{i}^{2}\right)\right)^{2}}\) & MN & [–100,100] & \(f(\overrightarrow{0})=0\) \\
**Rastrigin** & \(f_{8}(x)=\sum\limits_{i=1}^{D}\left(x_{i}^{2}-10\cos(2\pi x_{i})+10\right)\) & MS & [–5.12,5.12] & \(f(\overrightarrow{0})=0\) \\
**Griewank** & \(f_{9}(x)=\frac{1}{400}\sum\limits_{i=1}^{D}x_{i}^{2}-\prod\limits_{i=1}^{D} \cos\left(\frac{x_{i}}{\sqrt{i}}\right)+1\) & MN & [–600,600] & \(f(\overrightarrow{0})=0\) \\
**Ackley** & \(f_{10}(x)=20+e-20\exp\left(-0.2\sqrt{\frac{1}{D}\sum\limits_{i=1}^{D}x_{i}^{2}}\right)\) & MN & [–32.768,32.768] & \(f(\overrightarrow{0})=0\) \\  & \(-\exp\left(\frac{1}{D}\sum\limits_{i=1}^{D}\cos(2\pi x_{i})\right)\) & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The 10 benchmark functions for experimental study. As seen from the results, the GABCG algorithm has great advantage of convergence property and applicability compared to the ABC and GABC algorithms. For the unimodal functions in Figure 3(1-5), the performances of the GABCG algorithm are both superior to the ABC algorithm and the GABC algorithm, which verifies the good convergence accuracy and rate of the GABCG algorithm. In Figure 3(6-10), it is more obvious to see the advantage of the GABCG algorithm in the accuracy of results for multimodal functions, which testifies the good global optimization capability of the GABCG algorithm. From these iterative curves, it can be seen that the GABCG algorithm shows better convergence performance than the GABC algorithm in the latter iterations. Comparing the performance of the GABCG algorithm in unimodal functions and multimodal functions, it can be found that this algorithm is superior in solving most multi-peak problems. Especially on some complex problems, such as Rastrigin and Griewank functions, the results of both the ABC and GABC algorithms appear to be stagnant in the middle phases of evolution. However, the GABCG algorithm can keep finding better solutions during almost the entire search. This indicates that the GABCG algorithm successfully reduces unnecessary computation costs and determines the exact optimization paths by gradient acceleration.