## 3 The Hybrid Evolutionary Optimization Method Coupling CA with GAs
### Belief Space  
In this paper, according to the characteristics of genetic algorithms, combined with the manner of extracting and updating knowledge sources in the belief space, the knowledge sources are divided into situational knowledge, normative knowledge and historical knowledge. The manner of updating the knowledge sources in the belief space every \(K\) generations is adopted, so that the memory consumption brought by redundant information can be reduced. Different knowledge sources have different update strategies. Taking the maximization problem as an example, the update of the knowledge sources is described as follows:  
1. Situational knowledge. Situational knowledge was proposed by [PERSON] in 1997 [30] to record the excellent individuals with a guiding role in the evolutionary process and is structured as follows: \[<E_{1},E_{2},\ldots,E_{n}>\] (1) where \(E_{i}\) is the \(i\)th best individual, and in this paper the best individual is selected to update the situational knowledge, that is, \(s=1\). The process of updating situational knowledge is described as follows: \[<E(T+1)>=\begin{cases}<x_{b}>&\text{if }f(x_{b})>f(E(T))\\ <E(T)>&\text{else}\end{cases}\] (2) where \(x_{b}\) is the best individual in the Tth generation of the population space.
2. Normative knowledge. Normative knowledge was also proposed by [PERSON] [30] for limiting the search space and judging the feasibility of an individual. When an individual is outside the search space described by the normative knowledge, the normative knowledge will guide the individual into the dominant search space through the influence function, thus ensuring that evolution proceeds is in the dominant region, and for the \(n\)-dimensional optimization problem, the structure of the normative knowledge is described as follows: \[<V_{1},V_{2},\ldots,V_{n}>\] (3) where \(V_{i}=[(l_{i},u_{i}),(L_{i},U_{i})],i\leq n\). \(u_{i}\) and \(l_{i}\) are the upper and lower bounds of the \(i\)th dimensional variables, and \(U_{i}\) and \(L_{i}\) are the upper and lower bounds of the fitness value, respectively. The normative knowledge is updated with the change of the dominant search region, and gradually approaches the region where the best individual is located. Therefore, when there is a better individual in the \(T\)th generation beyond the current search range described by the normative knowledge, the normative knowledge is updated as follows: \[l_{i}(T+1)=\begin{cases}x_{j}^{i}(T)&\text{if }x_{j}^{i}(T)\leq l_{i}(T) \text{ or }f(x_{j}(T))>L_{i}(T)\\ l_{i}(T)&\text{otherwise}\end{cases}\] (4) \[L_{i}(T+1)=\begin{cases}f(x_{j}(T))&\text{if }x_{j}^{i}(T)\leq l_{i}(T) \text{ or }f(x_{j}(T))>L_{i}(T)\\ L_{i}(T)&\text{otherwise}\end{cases}\] (5) \[u_{i}(T+1)=\begin{cases}x_{j}^{i}(T)&\text{if }x_{j}^{i}(T)\geq u_{i}(T) \text{ or }f(x_{j}(T))>U_{i}(T)\\ u_{i}(T)&\text{otherwise}\end{cases}\] (6) \[U_{i}(T+1)=\begin{cases}f(x_{j}(T))&\text{if }x_{j}^{i}(T)\geq u_{i}(T) \text{ or }f(x_{j}(T))>U_{i}(T)\\ U_{i}(T)&\text{otherwise}\end{cases}\] (7)
3. Historical knowledge. Historical knowledge was introduced into the belief space by [PERSON] [31] to record important events that occurred during the evolutionary process, and its main role is to adjust the offset distance and direction when the optimization falls into a local optima. The historical knowledge structure is divided asshown in Figure 4, where \(e_{i}\) is the \(i\)th outstanding individual of historical knowledge preservation, \(W\) is its maximum capacity, and \(d_{sj}\) and \(d_{rj}\) are the average offset distance and the average offset direction of the \(j\)th design variable. The expressions of \(d_{sj}\) and \(d_{rj}\) are as follows:  
\[d_{sj}=\frac{\sum\limits_{k=1}^{\varpi-1}|e_{k+1}x_{j}-e_{k}x_{j}|}{w-1} \tag{8}\]  
\[d_{rj}=sgn\left(\sum\limits_{k=1}^{w-1}|e_{k+1}x_{j}-e_{k}x_{j}|\right) \tag{9}\]