# A Hybrid Optimization Framework with Dynamic Transition Scheme for Large-Scale Portfolio Management
## 3 Methodology
### Dynamic Information Transition

Table 2: _Cont._

\begin{table}
\begin{tabular}{c c c c c c c}

Figure 4: The Comparison of the Convergence on the CEC2019 Set. #### 4.1.3 The Analysis of the Computational Cost

Through analyzing the CPU time of all algorithms in Table 4, PSO runs faster than others on all evaluations, but it fails to get competitive solutions. As it unavoidably spends extra time to deal with the state and information transition when compared with the AMPO and CSDE approaches, the HYPO framework occupies more computing resources to reach the predefined generations. In fact, the HYPO algorithm can use a shorter time instead of running a whole search period to generate a good-performing solution when reviewing the convergence graphs. ### Portfolio Optimization

Portfolio optimization is still a challenging real-world problem in computational finance in which fund managers expect to maximize the returns while maintaining lower risk through adjusting the weights of assets in a portfolio. By formulating it as a combinatorial optimization problem, this section mainly focuses on the relationship between the return and risk in terms of the mean-variance model. In the experiments, there are three datasets from the constituents of different financial market indices, including CSI300, SP500, and a mixed market index (Mix for short) consisting of SP500, CSI300 and HSI. The testing period is 5 years from July 1, 2017 to June 30, 2022, and the stock will be removed from the dataset if it is not included in the market index during the whole testing period. The details of three datasets are described in Table 5 in which the risk-free rate comes from the average daily values of 5-year bond yield on the corresponding market. More importantly, to test the scalability of all algorithms, different scales of portfolios on the same dataset are selected and evaluated according to the ascending order of the market capitalization of companies. Besides, only long positions will be allowed during the trading in order to simplify the scenario. \begin{table}
\begin{tabular}{c c c c c c c c} \hline
**Func.** & **GA** & **PSO** & **WOA** & **DE** & **AMPO** & **CSDE** & **SADE** & **HYPO** \\ \hline F1 & 638.22 & **317.07** & 595.80 & 756.88 & 539.18 & 906.76 & 903.65 & 993.66 \\ F2 & 769.90 & **371.86** & 732.57 & 906.73 & 589.00 & 1047.10 & 1032.58 & 1107.29 \\ F3 & 787.34 & **378.54** & 763.77 & 918.48 & 589.40 & 1053.28 & 1048.54 & 1121.27 \\ F4 & 664.01 & **348.81** & 642.87 & 809.08 & 574.21 & 951.11 & 964.52 & 1037.94 \\ F5 & 842.39 & **408.55** & 798.04 & 1003.58 & 636.22 & 1096.68 & 1218.80 & 1188.41 \\ F6 & 799.28 & **428.90** & 778.36 & 961.12 & 646.37 & 1091.01 & 1212.15 & 1192.42 \\ F7 & 289.29 & **159.16** & 275.36 & 522.87 & 396.95 & 582.48 & 669.38 & 657.20 \\ F8 & 798.13 & **450.79** & 786.29 & 941.81 & 681.64 & 1057.11 & 1183.89 & 1179.93 \\ F9 & 957.95 & **515.62** & 938.90 & 1124.20 & 754.76 & 1220.32 & 854.17 & 1327.15 \\ F10 & 920.01 & **520.04** & 934.95 & 1079.48 & 760.35 & 1211.56 & 882.74 & 1257.85 \\ F11 & 747.12 & **439.60** & 745.37 & 910.93 & 679.18 & 1010.10 & 777.87 & 1053.56 \\ F12 & 109.19 & **57.13** & 97.17 & 225.75 & 289.52 & 386.91 & 381.35 & 408.64 \\ F13 & 764.05 & **440.75** & 743.65 & 941.31 & 666.62 & 1043.53 & 771.91 & 1059.93 \\ F14 & 741.46 & **432.56** & 749.63 & 896.82 & 657.37 & 1039.93 & 775.99 & 1055.90 \\ F15 & 560.72 & **295.09** & 514.90 & 707.74 & 518.65 & 825.21 & 627.32 & 857.73 \\ \hline \end{tabular}

Unit: CPU Second(s).

\end{table} 
Table 4: The CPU Time Comparison on the CEC2019 Set. \begin{table}
\begin{tabular}{c c c c} \hline
**Market Index** & **SP500** & **CSI300** & **Mix** \\ \hline Trading Days & 1258 & 1213 & 1148 \\ Risk-free Rate & 1.6575 & 3.0370 & 2.0042 \\ Number of Stocks & 490 & 239 & 788 \\ Testing Scale of & 50, 100, 150, 200, 239 & 100, 200, 300, 400, 490 & 100, 200, 400, 600, 788 \\ Portfolios & & & & \\ \end{tabular}
\end{table} 
Table 5: The Descriptions of Portfolio Optimization Datasets.