# A Brain Storm and Chaotic Accelerated Particle Swarm Optimization Hybridization
## 2 Background
### The BSO Algorithm
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Function** & **Formula** & **Global Minimum** & **Search Area** \\ \hline Sphere & \(f_{1}=\sum\limits_{i=1}^{D}x_{i}^{2}\) & \(f_{1}(\mathbf{x}^{*})=0\), & \([-100,100]^{D}\) \\ \(\mathbf{x}^{*}=\{0,\ldots,0\}\) & \([-100,100]^{D}\) \\ \hline Rosenbrock & \(f_{2}=\sum\limits_{i=1}^{D-1}[100(x_{i+1}-x_{i}^{2})^{2}+(1-x_{i})^{2}]\) & \(f_{2}(\mathbf{x}^{*})=0\), & \\  & \(\mathbf{x}^{*}=\{1,\ldots,1\}\) & \([-10,10]^{D}\) \\ \hline Schwefel 2.21 & \(f_{3}=\max\limits_{i=1,\ldots,D}|x_{i}|\) & \(f_{3}(\mathbf{x}^{*})=0\), & \([-100,100]^{D}\) \\ [PERSON] & \(f_{4}=10D+\sum\limits_{i=1}^{D}[x_{i}^{2}-10\cos(2\pi x_{i})]\) & \(f_{4}(\mathbf{x}^{*})=0\), & \([-5.12,5.12]^{D}\) \\  & \(\mathbf{x}^{*}=\{0,\ldots,0\}\) & \([-32.768,32.768]^{D}\) \\ \hline [PERSON] & \(f_{5}=-20\text{exp}\left(-0.2\sqrt{\frac{1}{D}\sum\limits_{n=1}^{D}x_{i}^{2}} \right)-\text{exp}\left(\frac{1}{D}\prod\limits_{i=1}^{D}\cos 2\pi x_{i}\right)+ \text{e}+20\) & \(f_{5}(\mathbf{x}^{*})=0\), & \\  & \(\mathbf{x}^{*}=\{0,\ldots,0\}\) & \([-32.768,32.768]^{D}\) \\ \hline Griewank & \(f_{6}=\frac{1}{4000}\sum\limits_{i=1}^{D}x_{i}^{2}-\prod\limits_{i=1}^{D}\cos \left(\frac{x_{i}}{\sqrt{i}}\right)+1\) & \(f_{6}(\mathbf{x}^{*})=0\), & \\  & \(\mathbf{x}^{*}=\{0,\ldots,0\}\) & \([-600,600]^{D}\) \\ \hline Alpine 1 & \(f_{7}=\sum\limits_{i=1}^{D}|x_{i}\sin\left(x_{i}\right)+0.1x_{i}|\) & \(f_{7}(\mathbf{x}^{*})=0\), & \([-100,100]^{D}\) \\  & & \(\mathbf{x}^{*}=\{0,\ldots,0\}\) & \([-100,100]^{D}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Benchmark functions. #### 4.2.1. BSO Parameters

Both for the stand-alone BSO and the BSO-CAPSO hybrid, \(p_{rep}=0.2\), \(p_{gen}=0.8\), \(p_{oneC}=0.4\) and \(p_{twoC}=0.5\), as suggested by BSO parameter selection studies [49]. #### 4.2.2. CAPSO Parameters

For \(\alpha(t)\), we used Equation (9). We set the parameter \(\gamma\) as \(\gamma=0.99\) in order to be similar and comparable to the parameters chosen for the BSO-CAPSO hybrid. #### 4.2.3. BSO-CAPSO Hybrid Parameters

For the CAPSO phase, the \(\beta\) parameter updated through Equation (10). Additionally, its values were normalized in \([0.2,0.7]\)[17]. For \(\alpha(t)\), we used Equation (9). Through experimentation, we noticed that values of \(\gamma\simeq 0.99\) were the most beneficial. To produce such values, we used the following formula (a similar approach can be found in Ref. [50]):

\[\gamma=(10^{-20})^{\frac{1}{c^{\intercal}}}, \tag{13}\]

where \(c>0\), for which values of \(c\in(1,7.5)\) were adequate. It is also recommended to increase and adjust \(\gamma\) using (13) if the dimensions of the optimization problem increase. The adjusted \(t_{switch}\) and the \(\gamma\) values per benchmark function \(f_{k}\) are shown in Table 2. We note that if \(t_{switch}\) was chosen somewhere between 2.5% and 20% of the number of iterations, \(T\), the hybrid provided adequate solutions. The following \(t_{switch}\) values were chosen with BSO's behavior and the optimization problem's complexity in mind. We note that parameter tuning was applied while considering performance, but also stability. In general, stochastic algorithms, such as the ones employed in this paper, have a chance of providing outlier results, or results of varying orders of magnitude; this could be seen in BSO's behavior for \(f_{6},f_{7}\), or the hybrid's results for \(f_{6}\), as presented in Table 3. During the process of parameter selection, the hybrid provided outlier-like results for some parameter sets ( \(t_{switch}\) and \(\gamma\)). This was more probable in multimodal functions or functions with broader search areas when the parameters were not adequate. The provided sets of parameters and results were chosen with algorithm stability in mind, and each set of 25 algorithm runs was handled as a unit, a nonseparable experiment. It is also noted that for an acceptably wide range of parameter sets, outlier results were seldom and comparable to typical stochastic/metaheuristic algorithm behavior.