# A Hybrid Optimization Framework with Dynamic Transition Scheme for Large-Scale Portfolio Management
## 2 Preliminaries
### The Adaptive Multi-Population Optimization Algorithm  
As originated from hybridized meta-heuristic algorithms, the Adaptive Multi-Population Optimization (AMPO) [28; 29; 30] is a newly meta-heuristic algorithm that integrates both EC and SI techniques. To flexibly adapt to more tasks at different search stages, the AMPO approach introduces five search groups, including local search group, global search group, random search group, leader group and migration group. Multiple populations are assigned to those groups for performing different search behaviors, respectively. Specifically, the members of each population are not fixed and will be transferred to other populations when triggering corresponding criteria. Among these search groups, the global search group and local search group imitate the swarm intelligence mechanisms conducting the search toward the currently best candidate or around themselves whereas the migration group performs the DE/rand/1/bin search. For the remaining search groups, the random group tries to distribute initial candidates throughout the search space to escape from local minima while the leader group always keeps the best solution for the current time. In the AMPO framework, the update rule of candidates in the global search group is shown as Equations (1) and (2):  
\[S_{i,t}^{S}=\omega\times S_{i,t-1}^{S}+\beta\times(gbest_{t-1}-x_{i,t-1}), \tag{1}\]  
\[x_{i,t}=x_{i,t-1}+S_{i,t}^{S}, \tag{2}\]  
where \(x_{i,t}\) is the solution of \(i^{th}\) candidate at iteration \(t\), \(S_{i,t}^{S}\) is the global step size, \(\omega\in(0,1)\) is the constant decay rate of global step size, \(\beta\in U(0,1)\) is the control factor generated by uniform distribution, and \(gbest_{t-1}\) is the best solution at the last iteration. Furthermore, the new solution of candidates in the local search group is described as Equation (3) to Equation (5):  
\[\sigma_{i,t}=\gamma\times\sigma_{i,t-1}, \tag{3}\]  
\[S_{i,t}^{l}=N(0,\sigma_{i,t})\times x_{i,t-1}, \tag{4}\]  
\[x_{i,t}=x_{i,t-1}+S_{i,t}^{l}, \tag{5}\]  
where \(x_{i,t}\) is the solution of \(i^{th}\) candidate at iteration \(t\), \(S_{i,t}^{l}\) is the local step size, \(N\) is the Gaussian random number generator with the mean value at 0 and standard deviation \(\sigma_{i,t}\), and \(\gamma\in(0,1)\) is the constant decay rate of \(\sigma_{i,t-1}\). As clearly depicted in Figure 1, the AMPO algorithm follows an evolution-like workflow. At the beginning of the search, all user-defined hyper-parameters are inputted into the algorithm, and new candidates are initialized. In the transformation stage, all candidates in the random group are transferred to one of the global or local search groups by preset transformation probabilities, and then those candidates fully or partially inherit the solution and individual parameters from candidates in the source group where the global search group, local search group and leader group are included. This will encourage candidates to explore promising areas and reduce meaningless random searches at the later stage. After that, all candidates update the control parameters and conduct the search based on rules that are defined by their corresponding search group. The candidates in the global search group will gradually approach the best current candidate from different directions. Meanwhile, the individuals in the local search group walk around the locally best solutions to jump out the local minima. Besides, the exploration ability of the AMPO will be weakened when all individuals in the random search group are transferred to other groups after iterations. Thus, the recovery mechanism is set up to reset part of the poor-performing candidates in the source group to the global search group or random search group. The updated candidates are evaluated and sorted, and then the currently best solution will be selected to replace the candidate in the leader group.