## 2 Short Introduction to Pareto Optimization and Two Aggregation Methods
### Pareto Optimization  
A decision vector \(x\in S\)dominates another vector \(y\in S\), if  
\[\forall\,i\in\bigl{\{}1,2,\ldots,k\bigr{\}}:\,f_{i}\bigl{(}x \bigr{)}\geq f_{i}\bigl{(}y\bigr{)}\ \ \text{and}\] \[\exists\,j\in\bigl{\{}1,2,\ldots,k\bigr{\}}:\,f_{j}\bigl{(}x \bigr{)}>f_{j}\bigl{(}y\bigr{)} \tag{2}\]  
A decision vector \(x^{\prime}\in S\), which is not dominated by any other \(x\in S\), is called _Pareto optimal_. The objective vector \(z^{\prime}=f\bigl{(}x^{\prime}\bigr{)}\) is Pareto optimal, if the corresponding decision vector is Pareto optimal and the corresponding sets can be denoted by \(P(S)\) and \(P(Z)\). The set of weakly Pareto optimal solutions, which is a superset of the set of Pareto optimal solutions, is formed by decision vectors, for which the following applies: An \(x^{\prime}\in S\) is called _weakly Pareto optimal_, if no other \(x\in S\) exists such that \(f_{i}\bigl{(}x\bigr{)}>f_{i}\bigl{(}x^{\prime}\bigr{)}\) for all \(i=1,\,\ldots,\,k\). As the set of Pareto optimal solutions consists of decision vectors only, which are not dominated, they can be regarded as the set of good compromises mentioned in the introduction. It follows from the definition that they are located on the border of the feasible objective region, as shown in the right part of Figure 1. The figure also illustrates the concept of weakly Pareto optimal solutions lying on the part of the green line outside of the section bounded by the black circles  
Figure 1: Feasible region \(S\) and its image, the feasible objective region \(Z\) for \(n=k=2\). The set of weakly Pareto optimal solutions is shown as a bold green line in the diagram on the right. The subset of Pareto optimal solutions is the part of the green line between the black circles. The ideal objective vector \(z^{\text{*}}\) consists of the upper bounds of the Pareto set.  
in the given example. It should be stated that the set of Pareto optimal solutions does not need to be as nicely shaped as shown in Figure 1; it may also be non-convex and disconnected.  
The upper bounds of the Pareto optimal set can be obtained by maximizing the \(f_{i}\) individually with respect to the feasible region. This results in the _ideal objective vector_\(z^{\ast}\!\in\mathfrak{R}^{k}\), an example of which is shown for the two-dimensional case in the right part of Figure 1. The lower bounds are usually hard to determine, see [3]. Although Pareto-based search methods can provide valuable estimations of the ranges of the objectives for practical applications, they are not suited for an exact determination of their lower and upper bounds.  
According to [4], constraints in the objective space are handled as follows: A solution \(x\)_constrained-dominates_ a solution \(y\), if any of the three conditions is satisfied:  
* Solution \(x\) is feasible and \(y\) is not.
* Both solutions are feasible and \(x\) dominates \(y\).
* Both solutions are infeasible, but \(x\) has a smaller constrained violation than \(y\). If more than one constraint is violated, the violations are normalized, summed up, and compared.  
Hereinafter, the term _Pareto optimization_ is used for an optimization procedure employing Pareto optimality to assess and compare generated solutions.