# A New Chaotic-Based Approach for Multi-Objective Optimization
## The Tornado Chaotic Search Algorithm
### Tornado Principle
#### 2.2.3 Chaotic Fine Search (CFS)  
The proposed CFS procedure aims to speed up the intensification process and refines the accuracy of the search. Indeed, suppose that the solution \(X\) obtained at the end of \(CGS/CLS\) search processes is close to the global optimum \(X_{o}\) with precision \(10^{-p}\), \(p\in N\). That can be formulated as:  
\[X=X_{o}+\varepsilon,\quad\|\varepsilon\|<10^{-p} \tag{24}\]  
Then the challenge is how to go beyond the accuracy \(10^{-p}\)? One can observe that the distance \(\varepsilon\) can be interpreted as a parasitic signal of the solution, which is enough to filter in a suitable way to reach the global optimum, or it corresponds to the distance to which is the global optimum of its approximate solution. Thus, one solution is to conducts a search in a local area in which the radius adapts to the distance \(\varepsilon=X-X_{o}\), component by component. However, in practice, the global optimum is not known a priori. To overcome this difficulty, as we know that, as the search process proceeds the resulting solution \(X\) is supposed to be close enough to the global optimum, one solution is to consider instead of the relation (24) the difference between the current solution \(X\) and its decimals fractional parts of order \(\eta\), (\(\eta\in N\)):  
\[\varepsilon_{\eta}=|X-X_{\eta}|\]where \(X_{\eta}\) is the fractional of order \(\eta\), i.e., the closest point of X to the precision \(10^{-\eta}\) formalised as : \(X_{\eta}=10^{-\eta}round(10^{\eta}X)\) (see Figure 10). Furthermore, we propose to add a stochastic component in the round process in order to perturb a potential local optima. Indeed, we consider the stochastic round \([. ]_{st}\) defined by:  
\[[X]_{st}= \begin{cases}round(X)+P,&if\mod(k,2)=0\\ round(X),&otherwise\end{cases}  \tag{25}\]  
where \(P\sim U(-1,1)^{d}\) is a stochastic perturbation operated on \(X\) alternatively through The Tornado cycles. This way, the new formulation of the \(\eta-\)error of \(X\) is given by:  
\[\widetilde{\epsilon}_{\eta}(X)=|X-10^{-\eta}[10^{\eta}X]_{st})| \tag{26}\]  
The structure of the chaotic fine search CFS is similar to the CLS local chaotic search. Indeed, it proceeds by levelling approach creating \(N_{f}\) levels, except the fact that the local area of level \(\eta\) is defined by its radius \(\mathcal{R}_{\eta}\) based on the \(\eta-\)error \(\eta\) and given by:  
\[\mathcal{R}_{\eta}=\frac{1}{1+\eta^{2}}\widetilde{\epsilon}_{\eta},\quad \eta\in[\![0,N_{f}-1]\!] \tag{27}\]  
This way, the local area search in CFS is carried out in a narrow domain that allow a focus on the current solution adapted coordinate by coordinate unlike the uniform local search in CLS as illustrated in Figure 11. The modified radius \(\widetilde{R}_{\eta}\) is formulated as follows:  
\[\widetilde{R}_{\eta}=\left\{ \begin{array}{ll}s\times R\cdot\widetilde{ \epsilon}_{\eta},&if\quad r>0.5\\ T\cdot R\cdot\widetilde{\epsilon}_{\eta},&otherwise\end{array} \right. \tag{28}\]  
where \(r,s\sim U(0,1)\) and \(T\sim U(0,1)^{d}\). Figure 10: Illustration of the 10 power zoom via the successive fractional parts. The design of the radius \(\mathcal{R}_{\eta}\) allows to zoom at an exponential rate of decimal over the levels. Indeed, we have:  
\[\|\mathcal{R}_{\eta}\|\leqslant\|\vec{\varepsilon_{\eta}}\|.\mathcal{R}<10^{- \eta}\times\mathcal{R} \tag{29}\]  
As a consequence, the CFS provides an ultra fast exploitation of the neighborhood of the current solution and allows in principle the refinement of the optimum with a good precision. The Fine Chaotic Search (CFS) is described in Algorithm 4 and the Tornado algorithm is detailed in Algorithm 5:  
Figure 11: Illustration of the coordinate adaptative local search in CFS.