# A New Chaotic-Based Approach for Multi-Objective Optimization  
[PERSON]  
1 INRIA Lille--Nord Europe Parc Scientifique de la Haute Borne, 40, Avenue Halley, Bat A, 59650 Villeneuve d'ascq, France; [EMAIL_ADDRESS]  
2 LERMA Laboratory, Mohammadia school of engineering, Mohammed V University in Rabat, 10040 Rabat, Morocco; [EMAIL_ADDRESS]  
3 Correspondence: [EMAIL_ADDRESS]  
2  
[PERSON]  
1 INRIA Lille--Nord Europe Parc Scientifique de la Haute Borne, 40, Avenue Halley, Bat A, 59650 Villeneuve d'ascq, France; [EMAIL_ADDRESS]  
[PERSON]  
2 LERMA Laboratory, Mohammadia school of engineering, Mohammed V University in Rabat, 10040 Rabat, Morocco; [EMAIL_ADDRESS]  
Footnote 1: email: {Talbi, } {Tal  
**Definition 3** (Dominance). : _A vector \(\vec{u}=(u_{1},\ldots,u_{n})\) is said to dominate \(\vec{v}{=}(v_{1},\ldots,v_{n})\) (denoted by \(\vec{u}\preccurlyeq\vec{v}\)) if and only if \(\vec{u}\) is partially less than \(\vec{v}\), i.e., \(\forall i\in\{1,\ldots,n\}\), \(u_{i}\leq v_{i}\ \wedge\ \exists i\in\{1,\ldots,k\}\) : \(u_{i}<v_{i}\)._  
**Definition 4** (Pareto set). : _For a given MOP \(\vec{f}(\vec{x})\), the Pareto optimal set is defined as \(\mathcal{P}^{*}=\{\vec{x}\in X|{-}\exists\vec{x^{\prime}}\in X,\vec{f}(\vec{x ^{\prime}})\preccurlyeq\vec{f}(\vec{x})\}\)._  
**Definition 5** (Pareto front). : _For a given MOP \(\vec{f}(\vec{x})\) and its Pareto optimal set \(\mathcal{P}^{*}\), the Pareto front is defined as \(\mathcal{P}\mathcal{F}^{*}=\{\vec{f}(\vec{x}),\vec{x}\in\mathcal{P}^{*}\}\)._  
**Definition 6** (Reference point). : _A reference point \(z^{*}=[\mathbb{z}_{1},\mathbb{z}_{2},\ldots,\mathbb{z}_{n}]\) is a vector which defines the aspiration level (or goal) \(\overline{z}_{i}\) to reach for each objective \(f_{i}\)._  
**Definition 7** (Nadir point). : _A point \(y^{*}=(y^{*}_{1},y^{*}_{2},\ldots,y^{*}_{n})\) is the nadir point if it maximizes each objective function \(f_{i}\) of \(F\) over the Pareto set, i.e., \(y^{*}_{i}=max(f_{i}(x))\), \(x\in\mathcal{P}^{*},i\in[1,n]\)._  
The approaches developed for treating optimization problems can be mainly divided into deterministic and stochastic. Deterministic methods (e.g., linear programming, nonlinear programming, and mixed-integer nonlinear programming, etc.) provide a theoretical guarantee of locating the global optimum or at least a good approximation of it whereas the stochastic methods offer a guarantee in probability [1; 2; 3]. Most of the well-known metaheuristics (e.g., evolutionary algorithms, particle swarm, ant colonies) have been adapted to solve multi-objective problems [4; 5] with a growing number of applications [6; 7; 8]. Multi-objective metaheuristics can be classified in three main categories:  
* Scalarization-based approaches: this class of multi-objective metaheuristics contains the approaches which transform a MOP problem into a single-objective one or a set of such problems. Among these methods one can find the aggregation methods, weighted metrics, chebybycheff method, goal programming methods, achievement functions, goal attainment methods and the \(\epsilon\)-constraint methods [9; 10]. * Dominance-based approaches: the dominance-based approaches (Also named Pareto approaches.) use the concept of dominance and Pareto optimality to guide the search process. Since the beginning of the nineties, interest concerning MOPs area with Pareto approaches always grows. Most of Pareto approaches use EMO (Evolutionary Multi-criterion Optimization) algorithms. Population-based metaheuristics seem particularly suitable to solve MOPs, because they deal simultaneously with a set of solutions which allows to find several members of the Pareto optimal set in a single run of the algorithm. Moreover, they are less sensitive to the shape of the Pareto front (continuity, convexity). The main differences between the various proposed approaches arise in the following search components: fitness assignment, diversity management, and elitism [11].