# A Hybrid Optimization Framework with Dynamic Transition Scheme for Large-Scale Portfolio Management
## 3 Methodology
### Dynamic Information Transition
Then, to further discuss the impact of information types on transition, another version (HYPO-LocalBest for short) of the proposed framework is implemented in which the meta-heuristic approach transmits all good-performing solutions to replace the inferior solutions of another meta-heuristic algorithm. Table 9 demonstrates the results of different transition direction settings on some sampling functions from CEC2019 and portfolio optimization. The HYPO algorithm is ranked first on 5 out of the 7 functions while the Bi-AMPO-CSDE generates the best solutions on the remaining 2 functions. Compared to the algorithms with unidirectional transition or no transition, the outstanding performance of the HYPO approach reveals the benefits of sharing information between meta-heuristic algorithms for which the information from one side can provide a good reference to another meta-heuristic so that it can start the search from promising areas. Meanwhile, providing the extra information to another meta-heuristic branch can help to escape from local minima if the algorithm is stuck for a long time running by itself. Furthermore, the dynamic state transition scheme can suggest a good timing and direction for exchanging information than the unconditionally bi-directional transition

Figure 8: Visualization of Mean-variance Distribution on the Mixed Market Set. Figure 7: Visualization of Mean-variance Distribution on the CSI300 Set. as frequently exchanging information can reduce the diversification of a population and premature converge at the beginning of the search. In addition, what kind of information being transmitted is a concern when applying hybrid meta-heuristic mechanisms. Except for only transmitting the currently best solution of each meta-heuristic branch, another design that transmits all good-performing solutions from one side to replace the poor-performing solutions of the opposite side is evaluated and compared. As shown in Table 10, the HYPO-LocalBest outperforms the original HYPO on all four CEC2019 sampling functions whereas the original HYPO excels on the portfolio optimization problems. During the search, different individuals of a population may have locally optimal solutions of different subcomponents. More locally optimal solutions are transmitted and learnt by the other sides so that the HYPO-LocalBest scheme performs well on such large-scale non-separable functions like F7, F12 and F15 in which the decision variables are dependent upon each other and grouped by several subcomponents. ## 5 Conclusions

There are various meta-heuristic algorithms conducting different nature-inspired search mechanisms that may succeed in tackling certain optimization problems, yet possibly fail to solve other problems. Accordingly, hybridizing meta-heuristic algorithms into a \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Func.** & **Metric** & **HYPO-LocalBest** & **HYPO** \\ \hline F3 & Fit Mean & **2.05\(\times\)10\({}^{1}\)** & 2.05\(\times\)10\({}^{1}\) \\  & Std. & 1.00\(\times\)10\({}^{-1}\) & 7.74\(\times\)10\({}^{-2}\) \\ F7 & Fit Mean & **3.09\(\times\)10\({}^{6}\)** & 3.79\(\times\)10\({}^{6}\) \\  & Std. & 1.18\(\times\)10\({}^{6}\) & 1.40\(\times\)10\({}^{6}\) \\ F12 & Fit Mean & **1.29\(\times\)10\({}^{3}\)** & 1.52\(\times\)10\({}^{3}\) \\  & Std. & 1.39\(\times\)10\({}^{2}\) & 1.88\(\times\)10\({}^{2}\) \\ F15 & Fit Mean & **6.01\(\times\)10\({}^{6}\)** & 6.23\(\times\)10\({}^{6}\) \\  & Std. & 6.61\(\times\)10\({}^{5}\) & 3.96\(\times\)10\({}^{5}\) \\ SP500-490 Dim & SR Mean & 2.0128 & **2.0891** \\  & Std. & 4.81\(\times\)10\({}^{-2}\) & 1.82\(\times\)10\({}^{-3}\) \\ CSI300-239 Dim & SR Mean & 2.0546 & **2.0703** \\  & Std. & 1.35\(\times\)10\({}^{-2}\) & 1.48\(\times\)10\({}^{-4}\) \\ Mix-788 Dim & SR Mean & 2.7474 & **2.8804** \\  & Std. & 5.36\(\times\)10\({}^{-2}\) & 7.80\(\times\)10\({}^{-3}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 10: Comparative Results on Different Types of Information being Transmitted.