## 4 Mathematical Programming Relaxations
### Preprocessing Reductions and Dual Bounds
Indeed, it considers a minimum refueling and a maximal fuel consumption: the maximal fuel consumption is given by a bounds of maximal power and the minimum fuel level after refueling is at least the minimal refueling \(\mathbf{Rmin}_{i,k}\) with the positivity constraint CT11. Denoting \(W_{i,k}=\sum_{w}(1-d_{i,k,w})\) the week when outage \((i,k)\) begins, we have relations \(W_{i,k+1}\geqslant W_{i,k}+\mathbf{Da}_{i,k}+\mathbf{L}_{i,k}\). \(W_{i,k+1}\geqslant\widetilde{To}_{i,k}+\mathbf{Da}_{i,k}+\mathbf{L}_{i,k}\). This minoration is valid for all feasible solutions, taking the lower bound of the LHS induces: \(\widetilde{To}_{i,k+1}\geqslant\widetilde{To}_{i,k}+\mathbf{Da}_{i,k}+ \mathbf{L}_{i,k}\). (42) is thus a valid pre-processing to tighten values of \(\widetilde{To}_{i,k}\). From relations \(W_{i,k+1}\geqslant W_{i,k}+\mathbf{Da}_{i,k}+\mathbf{L}_{i,k}\), we also have \(\widetilde{Ta}_{i,k+1}\geqslant W_{i,k}+\mathbf{Da}_{i,k}+\mathbf{L}_{i,k}\) and then \(\widetilde{Ta}_{i,k+1}\geqslant\widetilde{To}_{i,k}+\mathbf{Da}_{i,k}+ \mathbf{L}_{i,k}\). (43) is thus a valid pre-processing to tighten values of \(\widetilde{Ta}_{i,k}\). **Proposition 4**. : _Denoting for all \((m,n)\in K\), \(\mathbf{q}_{m,n}=\prod_{l=m}^{n}\frac{\mathbf{Q}_{i,l}-1}{\mathbf{Q}_{i,l}}\), we have following relations which allow for delete variables \(x_{i,k,s^{\prime}}^{init}x_{i,k,s}^{fin}\):_  
\[x_{i,k,s}^{init}=\mathbf{q}_{1,k}\mathbf{Ai}_{i}+\sum_{l=0}^{k-1}\mathbf{q}_{ l+2,k}\left(r_{i,l+1}-\mathbf{q}_{l+1,l+1}\left(\sum_{t}\mathbf{F}_{t}\,p_{i,l,s,t}- \mathbf{Bo}_{i,l}\right)\right) \tag{44}\]  
\[x_{i,k,s}^{fin}=\mathbf{q}_{1,k}\mathbf{Ai}_{i}+\sum_{l=0}^{k-1}\mathbf{q}_{ l+2,k}\left(r_{i,l+1}-\mathbf{q}_{l+1,l+1}\left(\sum_{t}\mathbf{F}_{t}\,p_{i,l,s,t}- \mathbf{Bo}_{i,l}\right)\right)-\sum_{t}\mathbf{F}_{t}\,p_{i,k,s,t}^{\intercal 2} \tag{45}\]  
Proof. : We notice that Equations (10) and (11) induce a linear system of equalities considering \(x_{i,k,s^{\prime}}^{init}x_{i,k,s}^{fin}\) as variables. A recursion formula can be deduced for \(x_{i,k,s}^{init}\) where the initial condition is \(x_{i,0,s}^{init}=\mathbf{Ai}_{i}\):  
\[x_{i,k,s}^{init}-\mathbf{Bo}_{i,k}=r_{i,k}+\frac{\mathbf{Q}_{i,k}-1}{\mathbf{ Q}_{i,k}}(x_{i,k-1,s}^{init}-\sum_{t}\mathbf{F}^{t}\,p_{i,k-1,s,w}-\mathbf{Bo}_{ i,k-1})\]  
This is an induction formula \(x_{i,k,s}^{init}=a_{i,k-1,s}x_{i,k-1,s}^{init}+b_{i,k-1,s}\) with \(a_{i,k,s}=\frac{\mathbf{Q}_{i,k+1}-1}{\mathbf{Q}_{i,k+1}}\) and \(b_{i,k,s}=r_{i,k+1}-\frac{\mathbf{Q}_{i,k+1}-1}{\mathbf{Q}_{i,k+1}}\left(\sum_ {t}\mathbf{F}^{t}\,p_{i,k,s,t}^{\intercal 2}-\mathbf{Bo}_{i,k}\right)\). Lemma 1 (proven by induction in [23]) allows also to compute \(x_{i,k,s}^{init}\) as linear expressions of the variables \(r_{i,k^{\prime}}\) and \(p_{i,k^{\prime},s,t}\) for \(k^{\prime}<k\). \(x_{i,k,s}^{fin}\) are also linear expressions of the variables \(r_{i,k^{\prime}}\) and \(p_{i,k^{\prime},s,t}\) for \(k^{\prime}<k\). reporting last inequality in (10). **Lemma 1**. : _Let \((u_{n})_{n\in\mathbb{N}}\) be defined by induction with \(u_{n+1}=a_{n}u_{n}+b_{n}\), \(a_{n},b_{n}\in\mathbb{R}\). We have:_  
\[u_{n}=\left(\prod_{l=0}^{n-1}a_{l}\right)u_{o}+\sum_{l=0}^{n-1}\left(\prod_{m=l +1}^{n-1}a_{m}\right)b_{l} \tag{46}\]