## 4 Numerical Experiments
### Numerical Analysis  
In the numerical experiment, we try the number of initial populations in different orders of magnitude. Taking the efficiency of algorithm into account, we finally set the number of initial population to 100 in the experiment. Besides, we hold the best 10 populations in each iteration while the others carry on migrating and mutating. The model (28) and model (29) discusses the level of optimization of \(\varepsilon\)BBO algorithm under the SSD in low dimension. Form the result of model (28), under the constraint of \([0,0.6]\), the return of optimal portfolio by \(\varepsilon\)BBO is \(1.1736\%\), while the returns of portfolio by SA and LF are \(1.147\%\) and \(1.148\%\). Besides, the \(\varepsilon\)BBO is faster in running, but the number of iteration is much more than LF. Under the constraint of \([0,1]\), the return of optimal portfolio by \(\varepsilon\)BBO is \(1.1779\%\), while the returns of portfolio by GA are \(1.165\%\) and \(1.177\%\). The GA(100) has the same initial population number as \(\varepsilon\)BBO, while the efficiency of \(\varepsilon\)BBO is better than the GA. Under the constraint of \([-1,2]\), because the short-selling is allowed, the return of optimal portfolio gets greatly improved. Model (29) is an improvement to model (28) essentially, whose enhanced suitability function can weak the small transactions. Under the constraint of \([0,0.6]\), three kinds of algorithm get the similar portfolio, while the \(\varepsilon\)BBO is faster in running than SA and LF. Under the constraint of \([0,1]\), the return of optimal portfolio by \(\varepsilon\)BBO is \(1.1779\%\), which is better than the model (28). Under the constraint of \([-1,2]\), the return of optimal portfolio by \(\varepsilon\)BBO is \(1.3043\%\), while the returns of portfolio by SA and LF are \(1.23\%\) and \(1.26\%\). The \(\varepsilon\)BBO is also fast in running. In the comparison of model (28) and model (29), model (29) can approximate the optimal portfolio in less number of iteration than model (28). Because the model (29) is more complex, the running speed of model (29) is slower. However, the portfolio performance of model (29) is much better. Compared with the SA, the \(\varepsilon\)BBO has great advantages in all respects. Compared with the LF, the \(\varepsilon\)BBO requires large number of iteration to approximate the optimal portfolio due to its randomness. Therefore, the LF has a advantage of iterative procedure. Besides, the performance of GA is similar to LF. If having the same initial population, the efficiency of \(\varepsilon\)BBO is higher than GA. the \(\varepsilon\)BBO also has a strong ability to approximate the optimal portfolio. The model (30) discusses the effectiveness of \(\varepsilon\)BBO algorithm in high dimension. Under the constraint of \([0,0.1]\), the return of optimal portfolio by \(\varepsilon\)BBO is \(0.1325\%\), the number of asset whose investment proportion more than \(2\%\) is \(12\), the number of asset whose investment proportion more than \(1\%\) is \(34\), the number of asset whose investment proportion less than \(0.5\%\) is \(39\), and the maximum investment proportion of optimal portfolio is \(5.17\%\). Under the constraint of \([-0.05,0.15]\), the return of optimal portfolio by \(\varepsilon\)BBO is \(0.3197\%\), the number of asset whose investment proportion more than \(4\%\) is \(12\), the number of asset whose investment proportion more than \(2\%\) is \(35\), the number of short-selling asset is \(29\), the maximum investment proportion of optimal portfolio is \(7.97\%\) and the minimum investment proportion of optimal portfolio is \(-4.79\%\). As a comparison, the return of FTSE100 Index portfolio is \(0.0937\%\), the number of asset whose investment proportion more than \(2\%\) is \(16\), and the maximum investment proportion of portfolio is \(7.3\%\). Compared with the FTSE100 Index, the optimal portfolio of \(\varepsilon\)BBO is inclined to the concentrated investment and avoid the small amount of investment.