## 2 Robust Optimization
 \tag{2}\]  
where \(I\) and \(\boldsymbol{u}\) are the lower and upper bound vectors of the variables \(x\). The introduction of the variable \(y_{j}\) is motivated when \(x_{j}\) can assume negative values (\(J_{i}<0\)), as explained below. The second addendum of the inequality constraint in Equation (2):  
\[\beta_{i}(\boldsymbol{x},U)=\max_{\eta\in U}\left\{\sum\limits_{j\in J_{i}} \eta_{ij}\hat{a}_{ij}y_{j}\right\} \tag{3}\]  
represents the worst-case term. In other words, it can be said that the \(ith\) constraint is protected by \(\beta_{i}(\boldsymbol{x},U)\) against the uncertainty of the parameters \(\tilde{a}_{ij}\). Hence, the robust counterpart is a bi-level problem, and its tractability is affected by the geometry of the uncertainty set U because the way in which the inner maximization in Equation (2) is eliminated depends on it. Several choices are available in literature: box, ellipsoidal and polyhedral sets (Figure 1). The box set corresponds to the maximum possible protection level and assumes that all parameters will take the worst possible value (worst case scenario). It is the most straightforward approach but also the most conservative, with the highest deterioration of the objective function (Figure 1a). To address the excessive conservatism of the box set, an ellipsoidal uncertainty set has been proposed on the observation that corners tend to be unlikely to happen (Figure 1b). However, due to the non-linearity of the model, it leads to computationally complex robust counterparts (Second Order Conic Problem). The polyhedral representation constitutes a compromise between the two previous models because it still allows controlling conservatism while preserving computational tractability [22]. The idea behind this model is that, even if every uncertain parameter can always assume the worst-case value, only a few of them does it simultaneously, and their number is controlled by the so-called _Uncertainty Budget_, \(\Gamma\) (Figure 1c). Because in practice uncertainties are generally bounded, and it is useless to consider parameters' values that exceed the uncertain space \(\Xi\), it is generally convenient defining uncertainty sets generated by combining the ellipsoidal or polyhedral set with the box set. In the paper, for limiting the computational burden, the \"box + polyhedral\" model has been adopted (Figure 1d). For \(\Gamma=1\), the polyhedron is exactly inscribed by the box, and the intersection between the two sets is exactly the polyhedron; on the contrary, when \(\Gamma=1\,J_{i}\,|\), the box is exactly inscribed in the polyhedron, and the intersection between them is exactly the box. With this kind of model, it is possible to show that the _ith_ constraint of the robust counterpart, Equation (2), is equivalent to the following formulation:  
\[\left\{ \begin{array}{l}\sum\limits_{j}a_{ij}x_{j}+\sum\limits_{j\in J_{i}}w_ {ij}+\Gamma_{i}z_{i}\leq b_{i}\\ z_{i}+w_{ij}\geq\hat{a}_{ij}x_{j}\,\ \ \forall j\in J_{i}\\ I\leq\mathbf{x}\leq\mathbf{b}\\ z_{i}\geq 0\,\ w_{ij}\geq 0\,\ \mathbf{y}\geq\mathbf{0}\end{array} \right. \tag{4}\]  
where the auxiliary variables \(z_{i}\) and \(w_{ij}\) are used to eliminate the inner maximization by using its dual formulation. This process requires resorting to the absolute value \(\left|\,x_{j}\,\right|\). If the variable is positive, the absolute value operator can be directly removed. Otherwise, it can be eliminated by introducing an additional auxiliary variable \(y_{j}\) and the constraint \(-y_{j}\leq x_{j}\leq y_{j}\):  
\[\left\{ \begin{array}{l}\sum\limits_{j}a_{ij}x_{j}+\sum\limits_{j\in J_{i}}w_ {ij}+\Gamma_{i}z_{i}\leq b_{i}\\ z_{i}+w_{ij}\geq\hat{a}_{ij}x_{j}\,\ \ \forall j\in J_{i}\\ -y_{j}\leq x_{j}\leq y_{j}\,\ \ \ \forall j\in J_{i}\\ I\leq\mathbf{x}\leq\mathbf{b}\\ z_{i}\geq 0\,\ \ \ w_{ij}\geq 0\,\ \ \mathbf{y}\geq\mathbf{0}\end{array} \right. \tag{5}\]  
_Probabilistic Guarantees in Robust Optimization_  
Differently to the stochastic optimization, RO does not provide explicit control on the risk of constraint violation.