# A New Chaotic-Based Approach for Multi-Objective Optimization
## The Tornado Chaotic Search Algorithm
### Tornado Principle
#### 2.2.2 Chaotic Local Search (CLS)  
The CLS procedure is designed to refine the search by exploiting the neighborhood of the solution \(\omega\) found by the chaotic global search CGS. In fact, the search process is conducted near the current solution \(\omega\) within a local search area \(\mathcal{S}_{\text{I}}\) whose radius is \(\mathcal{R}_{\text{I}}=r\times\mathcal{R}\) focused on \(\omega\) (see Figure 5a), where \(r\sim U(0,1)\) is a random parameter that corresponds to the reduction rate, and \(\mathcal{R}\) denotes the radius of the search zone \(\mathcal{S}=\prod\limits_{i=1}^{n}\left[l_{i},u_{i}\right]\) such as:  
\[\mathcal{R}=\frac{1}{2}(U-L)=\Big{(}\frac{1}{2}(u_{1}-l_{1}),\ldots,\frac{1}{2} (u_{n}-l_{n})\Big{)} \tag{14}\]  
The CLS procedure uses the following strategy to produce chaotic variables:  
* Like the CGS, the CLS also involves a levelling approach by creating \(N_{\text{I}}\) chaotic levels focused on \(\omega\). The local search process in each chaotic level \(\eta\in\llbracket 0,N_{\text{I}}-1\rrbracket\), is limited to a local area \(\mathcal{S}_{l,\eta}\) focused on \(\omega\) (see Figure 5b) characterized by its radius \(\mathcal{R}_{\eta}\) defined by the following: \[\mathcal{R}_{\eta}=\gamma_{\eta}\times\mathcal{R}_{l}=r\times\gamma_{\eta} \times\mathcal{R},\] (15)  
Figure 4: Generation of chaotic variables by the symmetrization approach in CGS. where \(\gamma_{\eta}\) is a decreasing parameter through levels which we have formulated in this work as follows:  
\[\gamma_{\eta}=\frac{10^{-2s\eta}}{1+\eta},\quad s\sim U(0,1) \tag{16}\]  
This levelling approach used by the CLS can be interpreted as a progressive zoom focus on the current solution \(\omega\) carried out through \(N_{\mathrm{I}}\) chaotic levels, and \(\gamma_{\eta}\) is the factor indicating the speed of this zoom process \((\gamma_{\eta}\diagdown 0)\) (see Figure 5b). Furthermore, by looking for potential solutions relatively far from the current solution CLS contributes also to the exploration of the decision space. Indeed, once the CGS provides an initial solution \(\omega\), the CLS intensifies the search around this solution, through several chaotic layers. After each CGS run, CLS carry out several cycles of local search (i.e., \(M_{\mathrm{I}}\)) This way, the CLS participates also to the exploration of neighboring regions by following the zoom dynamic through the CLS cycles as shown in Figure 6.  
Figure 5: Illustration of the CLS mechanism. Figure 6: Illustration of exploration aspect in CLS with the zoom dynamic produced by many local search cycles. Moreover, in each chaotic level \(\eta\), CLS creates two symmetric chaotic variables \(X_{1},X_{2}\) defined as follows (Figure 7):  
\[X_{1}=Z\times\mathcal{R}_{\eta},\quad X_{2}=(1-Z)\times\mathcal{R}_{\eta}= \mathcal{R}_{\eta}-X_{1}. \tag{17}\]  
By randomly choosing an index \(p\in\{1,..,n\}\) a stochastic decomposition of \(\mathbb{R}^{n}\) is built given by:  
\[\mathbb{R}^{n}=\mathcal{D}\oplus\mathcal{H},\ \ \mathcal{D}=\mathbb{R}\times e _{p},\ \ \mathcal{H}=\text{vect}(e_{i})_{i\
eq p}. \tag{18}\]  
Based on this, a decomposition of each chaotic variable \(X_{i,(i=1,2)}\) is applied:  
\[X_{i}=X_{i,d}+X_{i,h}. \tag{19}\]  
Finally, from each chaotic variable \(X_{i,\ (i=1,2)}\), \(N_{p}\) symmetric chaotic points \(\left(X_{i,j}\right)_{1\leq j\leq N_{p}}\) are generated using the polygonal model (Figure 8): polygonal model  
\[\mathcal{R}_{\eta}=\gamma_{\eta}\times\mathcal{R}_{l}=r\times\gamma_{\eta} \times\mathcal{R},\]0  
Moreover, if \(\omega\) is close enough to the borders of the search area \(\mathcal{S}\), the search process risks to leave it and then may give an infeasible solution localized outside \(\mathcal{S}\).