# A Hybrid Optimization Framework with Dynamic Transition Scheme for Large-Scale Portfolio Management
## 3 Methodology
### Dynamic Information Transition
#### 4.2.2 Visualization of Mean-Variance

As clearly depicted in Figures 6-8, the mean-variance distribution of each algorithm is described. Among this, the ideal portfolio should appear in the lower right corner of the scatter graph. The portfolios generated by the GA and PSO methods are located in the lower left corner in which those algorithms tend towards a lower risk investment, but the returns are also less than others. In addition, although the portfolios collected by the WOA have higher returns, they have to undertake much higher risk and demonstrate unstable investment strategies. Compared with the DE, SADE, AMPO, and CSDE methods, the HYPO framework generates competitive portfolios in small-scale scenarios. However, when the portfolio scale becomes larger, the HYPO method significantly obtains higher returns at the same risk level. This can verify the idea that integrating multiple meta-heuristic mechanisms by using the dynamic transition scheme can achieve better performance than a single meta-heuristic algorithm for which the components with different natures can be optimized at the same time to avoid local minima in the HYPO framework. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline
**Dim.** & **Metric** & **GA** & **PSO** & **WOA** & **DE** & **AMPO** & **CSDE** & **SADE** & **HYPO** \\ \hline
100 & SR Mean & 1.7838 & 1.4992 & 1.8361 & 2.1323 & 2.2099 & 2.2112 & **2.2161** & 2.2161 \\  & SR Std. & 7.76\(\times 10^{-3}\) & 3.44\(\times 10^{-2}\) & 1.08\(\times 10^{-1}\) & 1.75\(\times 10^{-2}\) & 7.41\(\times 10^{-3}\) & 3.80\(\times 10^{-3}\) & 1.54\(\times 10^{-5}\) & 2.48\(\times 10^{-5}\) \\  & +/=/- & + & + & + & + & + & + & - & \\
200 & SR Mean & 1.6931 & 1.4900 & 1.7899 & 2.0852 & 2.3561 & 2.3578 & 2.4133 & **2.4285** \\  & SR Std. & 5.76\(\times 10^{-3}\) & 3.42\(\times 10^{-2}\) & 1.40\(\times 10^{-1}\) & 3.55\(\times 10^{-2}\) & 6.33\(\times 10^{-2}\) & 2.08\(\times 10^{-2}\) & 1.59\(\times 10^{-2}\) & 1.00\(\times 10^{-4}\) \\  & +/=/- & + & + & + & + & + & + & + & \\
400 & SR Mean & 1.5950 & 1.3624 & 1.8090 & 1.9796 & 2.5497 & 2.5251 & 2.6659 & **2.8473** \\  & SR Std. & 5.04\(\times 10^{-3}\) & 3.03\(\times 10^{-2}\) & 2.67\(\times 10^{-1}\) & 2.66\(\times 10^{-2}\) & 1.65\(\times 10^{-1}\) & 5.66\(\times 10^{-2}\) & 5.38\(\times 10^{-2}\) & 1.47\(\times 10^{-3}\) \\  & +/=/- & + & + & + & + & + & + & + & \\
600 & SR Mean & 1.5031 & 1.2905 & 1.7487 & 1.8034 & 2.4544 & 2.3837 & 2.4838 & **2.8804** \\  & SR Std. & 4.88\(\times 10^{-3}\) & 1.85\(\times 10^{-2}\) & 1.85\(\times 10^{-1}\) & 3.27\(\times 10^{-2}\) & 1.21\(\times 10^{-1}\) & 5.45\(\times 10^{-2}\) & 3.65\(\times 10^{-2}\) & 1.96\(\times 10^{-3}\) \\  & +/=/- & + & + & + & + & + & + & + & \\
788 & SR Mean & 1.3624 & 1.1627 & 1.6372 & 1.6277 & 2.2765 & 2.2180 & 2.2909 & **2.8804** \\  & SR Std. & 3.09\(\times 10^{-3}\) & 1.80\(\times 10^{-2}\) & 2.50\(\times 10^{-1}\) & 3.10\(\times 10^{-2}\) & 1.25\(\times 10^{-1}\) & 6.00\(\times 10^{-2}\) & 4.35\(\times 10^{-2}\) & 7.80\(\times 10^{-3}\) \\  & +/=/- & + & + & + & + & + & + & + & \\ \hline \end{tabular}
\end{table} 
Table 8: Comparative Results on the Mixed Market Set. Figure 6: Visualization of Mean-variance Distribution on the SP500 Set. ### Ablation Studies

To investigate the effectiveness of the proposed transition scheme for hybridizing meta-heuristics, a set of ablation experiments are conducted to study the significance of transition directions and the kinds of information being transmitted. First, four different settings of transition directions including the unconditionally unidirectional transition from the AMPO to the CSDE (named Uni-AMPO-CSDE), unconditionally unidirectional transition from the CSDE to the AMPO (named Uni-CSDE-AMPO), unconditionally bi-directional transition between the AMPO and CSDE (named Bi-AMPO-CSDE), and no information exchange between the two algorithms (named No-exchange) are evaluated and compared with the proposed framework. None of these schemes include the dynamic transition scheme.