# A New Chaotic-Based Approach for Multi-Objective Optimization
## The Tornado Chaotic Search Algorithm
### Tornado Principle
#### 2.2.1. Chaotic Global Search (CGS)  
CGS starts by mapping the chaotic sequence generated by the adopted standard Henon map variable \(Z\) into ranges of design variable \(X\) by considering the following transformations (Figure 1):  
\[X_{1}=L+Z(U-L).\quad X_{2}=\theta+Z(U-\theta),\quad X_{3}=U-Z(U-\theta),\quad \theta=\frac{1}{2}(L+U). \tag{3}\]  
Those proposed transformations aim to overcome the inherent rigidity of the chaos dynamics. * _Levelling approach:_ In order to provide more diversification in the chaotic distribution, CGS proceeds by levelling with \(N_{c}\) chaotic levels. In each chaotic level \(l\in\llbracket 1,N_{c}\rrbracket\), and for each iteration \(k\) three chaotic variables are generated through the following formulas: \[X_{1} =L+(U-L)\times Z_{k+(l-1)N_{c}}\] (4) \[X_{2} =\theta+(U-\theta)\times Z_{k+(l-1)N_{c}}\] (5) \[X_{3} =U-(U-\theta)\times Z_{k+(l-1)N_{c}}\] (6) Note that for sake of simplicity, henceforth \(X_{i,k}\) will be simply noted \(X_{i}\). * _Symmetrization approach:_ In high-dimensional space, the exploration of all the dimensions is not practical because of combinatorial explosion. To get around this difficulty, we have introduced a new strategy based on a stochastic decomposition of the search space \(\mathbb{R}^{n}\) into two vectorial subspaces: a vectorial line \(\mathcal{D}\) and its corresponding hyperplane \(\mathcal{H}\): \[\mathbb{R}^{n}=\mathcal{D}\oplus\mathcal{H},\ \ \mathcal{D}=\mathbb{R}\times e_{p},\ \ \mathcal{H}=\text{vect}(e_{i})_{i\
eq p}.\] (7) By consequence, \[\forall\,X=(x_{1},x_{2},\ldots,x_{n})\in\mathbb{R}^{n}\colon\ \ X=X_{d}+X_{h},\] (8) where \[X_{d}=(0,\cdot\cdot,0,x_{p},0,\cdot\cdot,0)\in\mathcal{D},X_{h}=(x_{1},\cdot \cdot,x_{p-1},0,x_{p+1},\cdot\cdot,x_{n})\in\mathcal{H}.\] (9)  
The symmetrization approach based on this stochastic decomposition of the design space offers two significant advantages:  
Figure 1: Selection of chaotic variables for CGS. Significant reduction of complexity in the high dimensional problem in a way as if we were dealing with a 2D space with four directions. * The symmetric chaos is more regular and more ergodic than the basic one (Figure 2). Thanks to the stochastic decomposition (Equation (7)), CGS generates four symmetric chaotic points using axial symmetries \(\mathcal{S}_{\theta+\mathcal{D}}\), \(\mathcal{S}_{\theta+\mathcal{H}}\) (Figure 3):  
\[ \begin{array}{llll}X_{i,1}&=X_{i},&X_{i,2}&=\mathcal{S}_{\theta+\mathcal{D}}( X_{i,1}),\\ X_{i,3}&=\mathcal{S}_{\theta+\mathcal{H}}(X_{i,2}),&X_{i,4}&=\mathcal{S}_{ \theta+\mathcal{D}}(X_{i,3})=\mathcal{S}_{\theta+\mathcal{H}}(X_{i,1}),\end{array}  \tag{10}\]  
where the axial symmetries \(\mathcal{S}_{\theta+\mathcal{D}}\), \(\mathcal{S}_{\theta+\mathcal{H}}\) are defined as follows:  
\[\mathcal{S}_{\theta+\mathcal{D}}(X) =X_{d}+(2\theta_{h}-X_{h}) \tag{11}\] \[\mathcal{S}_{\theta+\mathcal{H}}(X) =(2\theta_{d}-X_{d})+X_{h} \tag{12}\]  
In other words, \(\forall i\in\{1,2,3\}\):  
\[ \begin{array}{llll}X_{i,1}&=X_{i}=X_{i,d}+X_{i,h},&X_{i,2}&=X_{i,d}+2\theta_ {h}-X_{i,h},\\ X_{i,3}&=2\theta-X_{i,1},&X_{i,4}&=2\theta-X_{i,2}.\end{array}  \tag{13}\]  
At last, the best solution among these all generated chaotic points is selected as illustrated by Figure 4.  
Figure 3: Illustration of axial symmetries \(\mathcal{S}_{\theta+\mathcal{D}}\) and \(\mathcal{S}_{\theta+\mathcal{H}}\). Figure 2: Illustration of symmetrisation approach in 2D. The code of CGS is detailed in Algorithm 2.