# A Hybrid Optimization Framework with Dynamic Transition Scheme for Large-Scale Portfolio Management
## 3 Methodology
### Dynamic Information Transition
end if if(\(accDirection\geq T_{trigger}\)) and (\(currentDirection\
eq 0\))then /* Trigger bi-directional information transition */ \(accDirection\gets 1\). currentDirection \(\gets 0\). end if return\(currentDirection\),\(accDirection\),\(stateCount\). ```  
**Algorithm 4**The Mechanism of System State Transition  
```
Input:\(currentDirection\),\(contribSignal\),\(accDirection\),\(stateCount\),\(T_{Bi}\),\(T_{stable}\), \(T_{trigger}\). Output:\(currentDirection\),\(accDirection\),\(stateCount\). lastDirection \(\leftarrow\)\(currentDirection\). if(\(lastDirection=0\))then if(\(accDirection\geq T_{Bi}\))then /* Bi-directional information transition exits. */ \(accDirection\gets 1\). if(\(contribSignal=True\))then \(stateCount\gets 1\). currentDirection \(\leftarrow\)\(currentDirection\gets 1\). end if else if(\(contribSignal=True\))then \(stateCount\leftarrow\)\(stateCount+1\). if(\(stateCount=0\))then /* State transition from \(M_{C,1}\) to \(M_{A,1}\)*/ \(stateCount\gets 1\). currentDirection \(\gets 1\). end if else \(stateCount\leftarrow\)\(stateCount-1\). if(\(stateCount=0\))then /* State transition from \(M_{A,1}\) to \(M_{C,1}\)*/ \(stateCount\leftarrow-1\). currentDirection \(\leftarrow 2\). end if if(\(stateCount>T_{stable}\))then /* Move to the stable state \(S_{AMPO}\)*/ \(stateCount\gets T_{stable}\). end if if(\(stateCount<-T_{stable}\))then /* Move to the stable state \(S_{CSDE}\)*/ \(stateCount\leftarrow-T_{stable}\). end if  end if if(\(lastDirection=currentDirection\))then \(accDirection\leftarrow\)\(accDirection+1\). else \(accDirection\gets 1\). end if if(\(accDirection\geq T_{trigger}\)) and (\(currentDirection\
eq 0\))then /* Trigger bi-directional information transition */ \(accDirection\gets 1\). \(currentDirection\gets 0\). end if return\(currentDirection\),\(accDirection\),\(stateCount\). ```  
**Algorithm 5**The Algorithm Algorithm for Algorithm 5  
``` Input:\(currentDirection\),\(contribSignal\),\(accDirection\),\(stateCount\),\(T_{Bi}\),\(T_{stable}\), \(T_{trigger}\). Output:\(currentDirection\),\(accDirection\),\(stateCount\). lastDirection \(\leftarrow\)\(currentDirection\). if(\(lastDirection=0\))then \(accDirection\leftarrow\)\(accDirection\leftarrow\). end if if(\(accDirection\geq T_{trigger}\)) and (\(currentDirection\
eq 0\))then \(accDirection\leftarrow\). ## 4 Empirical Results and Discussion

To carefully evaluate the performance of the proposed hybrid framework against those of other meta-heuristic algorithms, including the Genetic Algorithm (GA) [17], Particle Swarm Optimization (PSO) [23], Whale Optimization Problem (WOA) [22], Differential Evolution (DE) [18], Self-adaptive Differential Evolution (SADE) [46], Adaptive Multi-Population Optimization (AMPO) [28], and CSDE [31], a set of popular large-scale optimization functions called CEC2019 [47] and a set of portfolio optimization problems of different dimensions are selected for testing the algorithms both on benchmark functions and real-world problems. In all experiments, the population size of each algorithm is set to 50 for a fair comparison. Besides, the user-defined parameters of compared approaches follow the literature [22; 28; 31; 46] that demonstrate the SOTA performance on testing cases. Among which the GA consisted of random tournament selection, perturbation mutation, and two-point crossover strategies while the DE algorithm applied the DE/Current-to-best/1 strategy. The detailed parameter settings of all compared algorithms are shown in Table 1. In addition, the Wilcoxon rank-sum test [48] is introduced to compare the statistical significance of the proposed framework against the compared approaches with a significance level at 0.05. More specifically, the symbols \"+\", \"=\" and \"-\" indicate that the results of the proposed hybrid framework are statistically better than, similar to, or worse than that of the compared algorithms, respectively. Lastly, all algorithms are implemented by Python on a desktop computer installed with the Intel i9-7900X processor running at 3.3 GHz and 64 GB of RAM.