# Adaptive Self-Scaling Brain-Storm Optimization via a Chaotic Search Mechanism
## 4 CLS Based on Adaptive Self-Scaling  
Chaos, as a characteristic of nonlinear dynamic systems, is characterized by boundedness, dynamic instability, pseudo-randomness, ergodicity, and aperiodicity and depends on the initial value and control parameters. Because of the ergodicity and randomicity, the chaotic system changes randomly. If the time period is long enough, the chaotic system eventually experiences every single state. Based on this characteristic, chaotic systems can be used to establish a search operator for optimizing the objective function. Though chaotic optimization works well in small search spaces, the optimization time in large search spaces is unacceptable [45]. Hence, chaotic searches are often integrated into other global optimization algorithms, e.g., the evolutionary algorithm, to improve their search ability. By substituting random sequences of control parameters into BSO, the above 12 chaotic maps can generate chaotic sequences in the local search process. In a study by our group [40], the local search strategy was proven to be an effective method to improve the search ability of BSO. However, traditional algorithms are likely to fall into local optima. Thus, we used the adaptive self-scaling mechanism to further improve the search ability of the BSO algorithm. The CLS with a single chaotic map is defined as follows:  
\[Z_{b}^{t}=Z_{c}^{t}+\mathrm{radius}(t)\cdot(U-L)*(y_{t}-0.5), \tag{15}\]  
where \(Z_{c}^{t}\) represents the position where the algorithm obtains the current optimal solution during the \(t\)-th iteration; \(Z_{b}^{t}\) denotes the position of the new individual generated from the current optimal solution through the CLS strategy; \(U\) and \(L\) stand for the upper and lower bounds of the search space, respectively, where \(y_{t}\) is a chaotic variable gained based on one of the 12 chaotic maps; and \(radius(t)\) refers to the chaotic search radius to control the search range, with \(radius(t)\in(0,1)\). The radius gradually decreases as the number of iterations increases, and the radius variation is defined as follows:  
\[\mathrm{radius}(t+1)=\gamma\cdot\mathrm{radius}(t), \tag{16}\]  
where \(\gamma\) is an adjustable coefficient that was set to 0.986 in this study, and the initial search radius \(radius(0)=0.03\). According to its definition, the CLS strategy is decided by the search radius. However, applying the same search radius scaling scheme for different optimization problems is not ideal. We developed an adaptive chaotic search strategy for different problems, in which the search range can be adaptively scaled:  
\[Z_{b}^{t}=Z_{c}^{t}+\left|Z_{i}^{t}-Z_{j}^{t}\right|\cdot(y_{t}-0.5), \tag{17}\]  
where \(Z_{i}^{t}\) and \(Z_{j}^{t}\) are two randomly selected individuals from the current population; that is, in each CLS process, the search range changes with the position of the current population. In the entire search process, if the new solution obtained by the improved CLS scheme outperforms the current optimal individual in terms of fitness, it replaces the current optimal individual. Otherwise, the current individual is kept until the next iteration. Thus, a single update operation can be performed according to the following formula:  
\[Z_{c}^{t+1}=\left\{ \begin{array}{ll}Z_{b^{\prime}}^{t}&\text{ if }Fit\big{(}Z_{b}^{t}\big{)}\leq Fit \big{(}Z_{c}^{t}\big{)}\\ Z_{c^{\prime}}^{t}&\text{ otherwise}.\end{array} \right. \tag{18}\]  
Based on the analysis and existing improvements in the BSO algorithm, we developed ABSO, which improves the search performance of the algorithm by adaptively adjusting the CLS radius. The specific steps of Algorithm 1 are as follows:In the experiment, we extended ABSO into two algorithms, ABSO-R and ABSO-P. In ABSO-R, we randomly selected one of the 12 chaotic map to perform the chaotic search.