## 2 Robust Optimization
Therefore, it is crucial to define a probabilistic guarantee for the robust solution that can be computed _a priori_ as a function of the structure and size of the uncertainty set. Indeed, the quality of an optimal solution relies on the definition of uncertainty sets that are guaranteed to satisfy a particular upper bound on the probability  
Figure 1: Uncertainty set U for a constraint with two uncertain parameters. of constrain violation: a tighter probabilistic bound allows the adoption of a smaller uncertainty set with the same guarantee of feasibility and, consequently, leads to a less conservative solution. Some theoretical _a priori_ upper bounds (\(\epsilon_{i}^{pri}\)) for the probability of constraint violation have been proposed in literature. One of the most common is given by:  
\[\epsilon_{i}^{prio}=e^{-\frac{\Delta_{i}^{2}}{2|i|}} \tag{6}\]  
where \(\Delta_{i}\) is the adjustable parameter of the specific uncertainty set adopted (e.g., \(\Gamma_{i}\) for the box + polyhedral set), and \(\mid\!\!\mid\!\mid\!i\!\mid\) is the total number of uncertain parameters in the _ith_ constraint [23]. While this bound is appealing for its simplicity, it is not so attractive for its over-conservatism, especially when the number of uncertain parameters in a given constraint and the acceptable risk are small. For instance, with the box + polyhedral uncertainty set, 6 uncertain parameters and an accepted violation constraint probability of 10%, the uncertainty budget \(\Gamma_{i}\) calculated from (6) is around 5.3 (Figure 2), almost equivalent to the box set representation, where all uncertain parameters are assumed at their worst extreme value (_worst-case_ scenario). The sole assumption used to derive the bound (6) is that the uncertain parameters are bounded and have known means. If the postulation of symmetric distribution is added, a better upper bound for a box + polyhedral uncertainty set has been derived in [24] and it is expressed as:  
\[\epsilon_{i}^{prio}=\frac{1}{2^{n}}\sum_{k=\left\lceil v\right\rceil}^{n} \left( \begin{array}{c}n\\ k\end{array} \right) \tag{7}\]  
where  
\[ \begin{array}{rcl}\
u&=&\frac{\Gamma_{i}+n}{2}\\ n&=&\left\{\begin{array}{c}|I_{i}|+1-\text{sgn}\;\Gamma_{i}\;\text{if}\; |\Gamma_{i}|+|I_{i}|\;\text{is even}\\ |I_{i}|-\text{sgn}\;\Gamma_{i}\;\;\text{if}\;|\Gamma_{i}|+|I_{i}|\;\text{is even} \end{array}\right.\end{array} \]  
As it can be seen from Figure 2, bound (7) is clearly tighter, so providing \"less conservative\" solutions. For the previous example, the uncertainty budget \(\Gamma_{i}\) is taken equal to 4.001, i.e., 4 of 6 uncertain parameters are allowed to assume their worst value and one can change by 0.001 of its extreme deviation to guarantee a risk of constraint violation that does not exceed 10% probability. A second approach for assessing the probabilistic guarantees is based on the solution of the RO problem (**x**), and for this reason the resulting bound is called _a posteriori_ bound (\(\epsilon_{i}^{post}\)). It is typically used for checking the probability of constraint violation of the results obtained with specific uncertainty sets defined by a priori probabilistic bounds. Indeed, a posteriori bound will provide probabilities that are as tight as or tighter than those provided by the a priori bound, but only when they are \"analogous\", i.e., the two bounds have to  
Figure 2: Comparison of probability bounds. be derived with the same procedure and based on the same assumptions. For the a priori bound expressed in Equation (7), two ways exist to extend its principles to a posteriori bound. The analogue a posteriori bound of (7) is:  
\[\varepsilon_{i}^{post}=\frac{1}{2^{n_{a}}}\sum_{k=\left\lceil v_{a}\right\rceil} ^{n_{a}}\left( \begin{array}{c}n_{a}\\ k\end{array} \right) \tag{8}\]  
where  
\[