# A Hybrid Optimization Framework with Dynamic Transition Scheme for Large-Scale Portfolio Management
## 2 Preliminaries
### The Differential Evolution with Combined Strategy  
DE has been proved the outstanding performance on many practical problems over the past few decades. The key to success is the trial vector generation strategies (i.e., mutation strategies) that utilize the difference between candidates to update the solutions. Through taking different individuals as the reference, mutation strategies perform different search patterns, some of which conduct the exploration while others run the exploitation. In fact, both exploration and exploitation are required for tackling a real-world problem at different search stages; thus, how to balance them during the whole search process has become a critical challenge in recent years. In addition, it is also necessary to consider the control factors of mutation and crossover strategies. A larger control factor can speed up the optimization process at the beginning of the search, but it may fail to converge later. Hence, similar to the learning rate in deep learning approaches, a flexible control factor can enable the algorithm to adapt more to different environments. CSDE [31] is a simple yet powerful adaptive DE algorithm in which the adaptability is achieved from two aspects including the selection of mutation strategy and the adjustment of control parameters. As inspired by the JADE [19], the CSDE approach proposes two new trial vector generation strategies.  
Figure 1: The Flowchart of the AMPO Approach.  
The DE/current-to-pbest/1 strategy encourages individuals to explore the neighborhood of themselves. The update rule is shown as Equation (6):  
\[v_{i,t}^{C}=x_{i,t}+F_{i,t}^{C}\times\left(x_{pbest,t}-x_{i,t}\right)+F_{i,t}^{C} \times\left(x_{r1,t}-x_{r2,t}\right), \tag{6}\]  
\[F_{i,t}^{C}=M_{t}\times M_{t}+(1-M_{t})\times I_{i,t}, \tag{7}\]  
where \(v_{i,t}^{C}\) is the trial vector of \(i^{th}\) individual at iteration \(t\) when using DE/current-to-pbest/1, \(x_{i,t}\) is the original individual, \(F_{i,t}^{C}\) is the control factor which is updated by Equation (7), \(x_{pbest,t}\) is one of the solutions randomly selected from \(topp\%\) individuals, \(x_{r1,t}\) and \(x_{r2,t}\) are the reference individuals, \(M_{t}\) is the individual independence macro-control factor, and \(I_{i,t}\) is the individual status.  
Another strategy named DE/pbest-to-rand/1 tries to exploit the promising areas around the locally best solutions. The vector generation strategy and control factor update function are presented as Equations (8) and (9):  
\[v_{i,t}^{R}=x_{pbest,t}+F_{i,t}^{R}\times\left(x_{r1,t}-x_{i,t}\right)+F_{i,t}^ {R}\times\left(x_{r2,t}-x_{i,t}\right), \tag{8}\]  
\[F_{i,t}^{R}=M_{t}\times M_{t}+(1-M_{t})\times P_{t}, \tag{9}\]  
where \(v_{i,t}^{R}\) is the trial vector of \(i^{th}\) individual at iteration \(t\) when using DE/pbest-to-rand/1, \(F_{i,t}^{R}\) is the control factor, and \(P_{t}\) is the modulo-based periodic parameter.  
Besides, it is worth noting that using \(topp\%\) individuals instead of the currently best individual as the reference can avoid the search trapping into local minima. Furthermore, to adapt the search strategy over the optimization process, the selection probability is introduced to choose mutation strategies by comparing the historical performance between the two strategies:  
\[SR_{t}=\frac{SR_{t-1}^{C}}{SR_{t-1}^{C}+SR_{t-1}^{R}}, \tag{10}\]  
where \(SR_{t-1}^{C}\) and \(SR_{t-1}^{R}\) are the success rate of the two proposed strategies, respectively.  
The workflow of the CSDE approach is demonstrated in Algorithm 1. It follows the usual operations of DE algorithms including mutation, crossover, evaluation, and selection, but the search strategies and control factors of each individual are dynamically adjusted at each iteration.