## 4 Numerical Experiments
### Example 1  
At first, we consider the portfolio optimization problem in low dimensional. We consider a history of percentage returns, for \(m=10\) time periods and a group of \(n=5\) assets in Table 1[9].  
Our aim is to find an optimal investment strategy for a fixed capital in n assets which maximizes the expected return. If we do not consider the impacts of transaction cost, based on Table 1 and model (19), we propose the following SSD constraints model:  
\[\begin{array}{ll}min&-g(x,\xi)\\ s.t.&\sum_{i=1}^{10}\frac{1}{10}(g(y,\xi^{j})-g(x,\xi^{i}))_{+} \leq\sum_{i=1}^{10}\frac{1}{10}(g(y,\xi^{j})-g(y,\xi^{i}))_{+},j=1,\ldots,10\\ &x\in X\end{array} \tag{28}\]  
where \(X\) has three conditions which we have discussed above. We apply the \(\varepsilon\)BBO algorithm to the model (28), and we let \(y=(0.2,0.2,0.2,0.2,0.2)\) as the reference group. Comparing the \(\varepsilon\)BBO algorithm with Genetic Algorithm (GA), stochastic approximation (SA) method and level function (LF) methods, we get the test results shown in Table 2[9; 28].  
The Figures 2-4 show the performance of the \(\varepsilon\)BBO and the optimal portfolio in three kinds of \(X\) constraints of model (28).  
In model (28) we do not consider the transaction cost, if we take it into account, in order to weaken the contribution of small transactions we propose the following model:  
\[\begin{array}{ll}min&-g(x,\xi)-\sum_{i=1}^{10}x_{i}^{2}\\ s.t.&\sum_{i=1}^{10}\frac{1}{10}(g(y,\xi^{j})-g(x,\xi^{i}))_{+} \leq\sum_{i=1}^{10}\frac{1}{10}(g(y,\xi^{j})-g(y,\xi^{i}))_{+},j=1,\ldots,10\\ &x\in X\end{array} \tag{29}\]  
The same as the model (28), we get the test results of model (29) shown in Table 3[9].  
\begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline  & & & & & & & & & & & & \\ \hline  & **1** & **2** & **3** & **4** & **5** & **6** & **7** & **8** & **9** & **10** \\ \hline Asset 1 & 1.2 & 1.3 & 1.4 & 1.5 & 1.1 & 1.2 & 1.1 & 1.0 & 1.0 & 1.1 \\ Asset 2 & 1.3 & 1.0 & 0.8 & 0.9 & 1.4 & 1.3 & 1.2 & 1.1 & 1.2 & 1.1 \\ Asset 3 & 0.9 & 1.1 & 1.0 & 1.1 & 1.1 & 1.3 & 1.2 & 1.1 & 1.0 & 1.1 \\ Asset 4 & 1.1 & 1.1 & 1.2 & 1.3 & 1.2 & 1.2 & 1.1 & 1.0 & 1.1 & 1.2 \\ Asset 5 & 0.80 & 0.75 & 0.65 & 0.75 & 0.80 & 0.90 & 1.00 & 1.10 & 1.10 & 1.20 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Rates of return on five assets over ten period.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**X Constraint** & **Algorithm** & **Iteration** & **Time** & **x** & **E[g(x,\(\xi\))]** \\ \hline  & \(\varepsilon\)BBO & 92 & 0.2833 & (0.595,0.005,0,4,0) & 1.1736 \\ \([0,0.6]\) & SA & 115 & 5 & (0.325,0.231,0.177,0.266,0) & 1.147 \\  & LF & 5 & 0.6355 & (0.325,0.231,0.177,0.266,0) & 1.148 \\  & \(\varepsilon\)BBO & 67 & 0.217 & (0.799,0.201,0.0) & 1.1779 \\ \([0,1]\) & GA(100) & 4 & 0.1322 & (0.576,0.203,0.027,0.188,0.006) & 1.165 \\  & GA(100) & 3 & 0.6536 & (0.75,0.175,0.075,0) & 1.177 \\ \([-1,2]\) & \(\varepsilon\)BBO & 59 & 0.25 & (0.097,0.468,0.49,0.701,-0.756) & 1.300 \\ \hline \hline \end{tabular}
\end{table}
Table 2: The model (28) using date in Table 1. Time is in minute and the expected return of the benchmark portfolio \(E[g(y,\xi)]=1.073\).  
Figure 4: The performance of the \(\varepsilon\)BBO and the optimal portfolio in \(X\in[-1,2]\) of model (28).  
Figure 3: The performance of the \(\varepsilon\)BBO and the optimal portfolio in \(X\in[0,1]\) of model (28).  
Figure 2: The performance of the \(\varepsilon\)BBO and the optimal portfolio in \(X\in[0,0.6]\) of model (28).