# A Hybrid Optimization Framework with Dynamic Transition Scheme for Large-Scale Portfolio Management
## 3 Methodology
### Dynamic Information Transition
### The CEC2019 Benchmark Functions

The CEC2019 benchmark set includes 15 minimization functions of 1000 dimensions, and then it can be further categorized into four different groups of interaction between variables. Specifically, \(F1\)-\(F3\) are fully separable functions while \(F4\)-\(F11\) are partially separable functions consisting of a set of non-separable subcomponents. Furthermore, \(F12\)-\(F14\) are overlapping functions in which some shared decision variables exist on two subcomponents. Lastly, \(F15\) is the fully non-separable function. To study the stability of all algorithms, there are 25 independent runs being conducted by each algorithm in which the mean and standard deviation of fitness values in all runs are saved. Meanwhile, the maximum number of function evaluations is set at \(3\times 10^{6}\). #### 4.1.1 Numerical Results

As shown in Table 2, the numerical results on the CEC2019 dataset reveal that the proposed HYPO framework achieves the 1 st rank on 11 out of the 15 large-scale benchmark functions, and most of them are non-sparable functions that are close to real-world applications. Besides, the CSDE algorithm performs the best on the rest of the problems in which half of them are fully separable functions. The details are discussed as follows in terms of different categories. \begin{table}
\begin{tabular}{c c} \hline \hline
**Algorithms** & **Parameter Settings** \\ \hline GA & \(\textit{TournamentSize}=3,\textit{PerturbationRate}=0.1\) \\ PSO & \(\omega=0.8,c_{1}=0.5,c_{2}=0.5\) \\ WOA & \(b=1\) \\ DE & \(F=0.5,CR=0.3\) \\ AMPO & \(P_{LS}^{LD}=0.8,P_{LS}^{LS}=0.8,PR=0.6,\omega=0.1\), \\ CSDE & \(\gamma=0.9\) \\ SADE & \(FP=200\) \\ SAPE & \(LP=40,\textit{F}_{mean}=0.5,\textit{F}_{std}=0.3\), \\ \(CR_{mean}=0.5,CR_{std}=0.1\) \\ HYPO & \(T_{Bi}=2,T_{stable}=2,\textit{T}_{trigger}=10\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The Parameter Settings of Compared Algorithms. * For the fully separable functions of \(F1\)-\(F3\), the HYPO algorithm attains relatively good solutions on \(F1\) and \(F3\) while keeping the stability at a low level on 25 runs. It is worth noting that the algorithms do not need to consider the correlation among variables for which all decision variables are independent. In fact, real-world problems are complicated and involve a large amount of dependent components. * For the partially separable functions of \(F4\)-\(F11\), \(F4\)-\(F7\) are the functions with a set of non-separable subcomponents and one fully separable subcomponent whereas \(F8\)-\(F11\) are the functions with only a set of non-separable subcomponents and no fully separable subcomponent. The proposed framework is ranked first on 3 out of 4 functions on two sub-categories evaluation, respectively. Specifically, the mean values on \(F7\), \(F8\) and \(F11\) collected by the HYPO approach are significantly smaller than that of compared algorithms. * For the overlapping functions of \(F12\)-\(F14\), the HYPO approach excels all compared algorithms on all functions in terms of both the mean and standard deviation of fitness values, especially on optimizing \(F13\) and \(F14\) in which mean values of the HYPO framework are at least an order of magnitude smaller than that of other algorithms. * For the fully non-separable function \(F15\), the proposed framework outperforms other compared algorithms. This clearly demonstrates that the HYPO algorithm has a strong capability to handle large-scale optimization problems with complicated dependent correlations among decision variables. Furthermore, Table 3 carefully counts the results of the significant test of each algorithm. It is evident that the proposed HYPO framework significantly achieves outstanding performance on solving large-scale problems when compared to other testing algorithms in the perspective of statistics, especially with the comparison of its two meta-heuristic components that are the AMPO and CSDE frameworks, for which the effectiveness of the contribution-based state transition scheme of the proposed framework can be verified.