## 2 Robust Optimization  
For typical real-world problems where system operation and components' responses may deviate from the ideal ones, decision-maker wishes to identify optimal solutions that protect himself against parameter variability and implementation uncertainty. This issue becomes of paramount importance when new technologies and innovative problem solutions are introduced, due to the lack of information on the on-field performances of the new devices or on the actual behaviour of the actors involved. Different mathematical approaches have been proposed in literature and applied into practice, such as Sensitivity Analysis, Stochastic Programming, Chance-Constrained methods, and Robust Optimization. The burgeoning success of RO in a wide selection of application areas lies in its computational tractability because the uncertainty model adopted is not stochastic, but rather deterministic and set-based. Instead of striving for a probabilistic immunization of the solution to stochastic uncertainty, it creates a solution that is feasible for any realization of the uncertainty in a given set. From this point of view, RO is a worst-case oriented methodology, and this aspect also determines its main drawback: the possible over-conservatism of the solution. Thus, the choice of the uncertain parameter set into which the worst-case is evaluated should be done to achieve an acceptable trade-off between system performance and protection against uncertainty, i.e., neither too small nor too large. For a Linear Programming (LP) optimization, the robust counterpart is written as:  
\[\begin{array}{l}\text{min}c^{T}x\\ \text{s.t. }:\quad\max_{\eta\in U}A(\eta)x\leq b\\  
where \(x\) is the vector of decision variables, \(A\) is a \(m\times n\) constraint matrix, \(b\in\mathbb{R}^{m}\) is the right-hand-side vector, \(\eta\) is a random variable, \(\Xi\) is the entire uncertain set and U is the uncertainty subset of \(\Xi\) used for the optimisation. The desired protection level against uncertainty depends on the extent of \(\Xi\) covered by U. For instance, if the goal is the maximum protection (no risk of constraints violation, i.e., worst-case scenario), then the robust optimisation has to consider simultaneously all possible variations of the input data and, accordingly, U \(\equiv\Xi\). However, this option is generally over-conservative, because it also considers combinations of parameters' values that are extremely rare to happen. Therefore, if a minimum risk is acceptable, a subset of \(\Xi\) can be used. In formulation (1), data uncertainty is supposed to affect only the elements of matrix \(A\), because the possible uncertainty in the objective function (\(\epsilon\) vector) can be easily moved among the inequality constraints by changing the objective function (\(\min z\)) and adding a new constraint (\(c^{T}x-z\leq 0\)). The generic uncertain coefficient \(\tilde{a}_{ij}\) of matrix \(A\) is modelled as a symmetric and bounded random variable that varies in the interval \([a_{ij}-\hat{a}_{ij},\,a_{ij}+\hat{a}_{ij}]\), where \(a_{ij}\) is the nominal value of \(\tilde{a}_{ij}\) and \(\tilde{a}_{ij}\) is the extreme deviation from the nominal value. The hypothesis of modelling uncertainty with symmetric and bounded random variables is necessary to preserve the convexity of U [21]. Associated to the uncertain coefficient \(\tilde{a}_{ij}\), it is defined the random variable \(\eta_{ij}=(\tilde{a}_{ij}-a_{ij})/\tilde{a}_{ij}\), which follows an unknown, but symmetric, distribution and takes values in [1, 1]. By indicating with \(J_{i}\) the set of uncertain coefficients (\(\tilde{a}_{ij}\)) in the \(ith\) row of the matrix \(A\), the generic inequality constraint of the LP problem (1) can be written as:  
\[ \begin{array}{l}\sum\limits_{j}a_{ij}x_{j}+\max_{\eta\in U}\left\{\sum \limits_{j\in J_{i}}\eta_{ij}\hat{a}_{ij}y_{j}\right\}\leq b_{i}\\ \text{with }-y_{j}\leq x_{j}\leq y_{j}\forall j\\ \boldsymbol{l}\leq\boldsymbol{x}\leq\boldsymbol{u}\\ \boldsymbol{y}\geq\boldsymbol{0}\end{array}