## 2 Background
### Multi-Objective Optimization  
Multi-objective optimization is the discipline dealing with problems formulated around two or more objective functions that have to be optimized at the same time [18; 19]. These functions are usually in conflict, so improving one of them leads to the degradation of the others. We assumed, without loss of generality, that all objective functions are to be minimized. Broadly speaking, the goal sought in a multi-objective optimization problem (MOP) is to find a set of solutions that not only satisfy eventually imposed equality and inequality constraints (as per the problem to be solved), but also best balance the trade-off between the established objectives, i.e., solutions for which it is not possible to find any others that improve (minimize) the values of the defined objectives. Mathematically:  
**Definition 1**. : _(MOP) Find a vector \(\vec{x}^{*}=\left[x_{1}^{*},x_{2}^{*},\ldots,x_{n}^{*}\right]\) that satisfies \(m\) inequality constraints \(g_{i}(\vec{x})\geq 0,i=1,2,\ldots,m\) and \(p\) equality constraints \(h_{i}(\vec{x})=0,i=1,2,\ldots,p\) (if any such constraints exists) and minimizes the vector function \(\vec{f}(\vec{x})=[f_{1}(\vec{x}),f_{2}(\vec{x}),\ldots,f_{k}(\vec{x})]^{T}\), where \(\vec{x}=[x_{1},x_{2},\ldots,x_{n}]^{T}\) is the vector of decision variables._In the above definition, all the values of the vector of decision variables that satisfy the constraints circumscribe the _feasible region_\(\Omega\), so any point \(\vec{x}\in\Omega\) can be referred to as a _feasible solution_. The conflicting nature of the objectives \(\vec{f}(\vec{x})\) of an MOP implies the need for defining what an optimal solution is. Connecting with the broad explanation of what an MOP pursues that is given above, this leads to the definition of _Pareto optimality_:  
Definition 2: (Pareto optimality) A point \(\vec{x}^{*}\in\Omega\) is said to be Pareto optimal if for every \(\vec{x}\in\Omega\) and \(1=\{1,2,\ldots,k\}\), either \(\forall_{i\in I}(f_{i}(\vec{x})=f_{i}(\vec{x}^{*}))\) or there is at least one \(i\in I\) such that \(f_{i}(\vec{x})>f_{i}(\vec{x}^{*})\). According to this definition, a vector (point or solution) \(\vec{x}^{*}\) is Pareto optimal if no feasible vector \(\vec{x}\) exists that would improve some criteria without causing a simultaneous worsening of at least one other criterion. Given two vectors corresponding to solutions of a given MOP, comparing them to determine which is better in terms of objective minimization is not as simple as in single-objective optimization. When dealing with a single objective, the vector having the lower associated function value can be declared to be better than the other. By contrast, when there are several objectives involved, the concept of _Pareto dominance_ is introduced to compare among solutions:  
Definition 3: (Pareto Dominance) A vector \(\vec{u}=(u_{1},\ldots,u_{k})\) is said to dominate \(\vec{v}\)=\((v_{1},\ldots,v_{k})\) (denoted by \(\vec{u}\preccurlyeq\vec{v}\)) if and only if \(\vec{u}\) is partially less than \(\vec{v}\), i.e., \(\forall i\in\{1,\ldots,k\},\ u_{i}\leq v_{i}\ \wedge\ \exists i\in\{1,\ldots,k\}:\ u_{i}<v_{i}\). By using Pareto dominance, we can determine whether, given two solutions, one _dominates_ the other, or instead, they do not dominate each other. In this last case, they are _non-dominated_ solutions, and therefore, it cannot be said which of them is better in the sense of the trade-off between the objectives of the MOP. Pareto optimality can be thus defined in terms of dominance, as a Pareto-optimal point is one that is not dominated by any other point in \(\Omega\). Optimizing an MOP can be defined then as the process of finding the set composed of all the Pareto-optimal solutions.