# A Matheuristic for Joint Optimal Power and Scheduling Assignment in DVB-T2 Networks +
## 3 Strengthening the Initial MILP Formulation
Here, we just concisely remind major well-known results about cover inequalities: if we are given a _knapsack constraint_\(\sum_{j\in J}a_{j}x_{j}\leq b\) with \(a_{j}\), \(b\in\mathbb{R}_{+}\) and \(x_{j}\in\{0,1\}\), \(\forall j\in J\), then we can replace it with its cover inequalities \(\sum_{j\in C}x_{j}\leq|C|-1\), where \(C\) is a _cover_. A cover represents a subset of indices \(C\subseteq J\) with the property that the sum of the corresponding coefficients \(a_{j},j\in C\) does not satisfy the knapsack constraint (i.e., \(\sum_{j\in C}a_{j}>b\)). Each cover inequality associated with a cover \(C\) can therefore be interpreted as a combination of binary variables \(x_{j}\) that cannot be set to \(1\) at the same time and thus we can activate at most \(|C|-1\) of them. The GUB cover inequalities represent a stronger version of the simple cover inequalities, which one can define exploiting the presence of GUB constraints. Proceeding as in [26; 33], we can define the general form of the GUB cover inequalities through which we can replace constraints in Equation (8), obtaining a stronger formulation:  
\[x_{ts}+\sum_{i\in\Sigma}\sum_{l=1}^{\lambda_{i}}z_{s\delta(\sigma,l) \downarrow}+\sum_{i\in\Gamma}\sum_{l=q_{i}}^{|L|}z_{i\delta(i,l)\downarrow} \leq|\Delta|+|\Gamma|\, \tag{9}\]  
in which \(\Sigma\subseteq S\) is a subset of stations emitting useful signals towards TP \(t\) and \(\Gamma\subseteq S\backslash\Sigma\) is a subset of interfering stations. Moreover, \((q_{1},\ldots,q_{|\Gamma|})\in L^{I}(t,\Sigma,\lambda,\Gamma)\), with \(L^{I}(t,\Sigma,\lambda,\Gamma)\subseteq L^{|\Gamma|}\) represent the subset of interfering levels of interfering stations in \(\Gamma\) that destroy the service of \(t\) provided by the subset of serving stations \(\Sigma\), emitting with power levels corresponding to indices \(\lambda=(\lambda_{1},\ldots,\lambda_{|\Sigma|})\). Intuitively, for a fixed TP, subset of serving stations \(\Sigma\) and subset of interfering stations \(\Gamma\), a GUB cover inequality is built by fixing a power setting of the serving stations and defining a power setting of the interfering stations that compromise the coverage of the considered TP. We remark that in what follows we do not completely replace the binary SIR constraints in Equation (9), but we add the GUB constraints that consider one single useful station and one single interferer to DVB-PI. Indeed, as shown in [26; 33], such a GUB provides a complete convex hull characterization for the single-server-single-interferer case and, even in the case of multiple serving stations and multiple interferers, still provide (tight) valid inequalities. We further strengthen the mathematical formulation by including additional valid inequalities that are aimed at modeling the presence of couples of SIR constraints that consider only two stations and that cannot be satisfied at the same time by any power configuration. In a more formal way, let us consider two SIR constraints corresponding with TPs \(t_{1}\) and \(t_{2}\) served by two stations \(s_{1}\) and \(s_{2}\):  
\[a_{t_{1}s_{1}}\cdot\left(\sum_{l\in L}p_{l}\cdot z_{s_{1}\delta( s_{1},t_{1})}\right)-\phi a_{t_{1}s_{2}}\cdot\left(\sum_{l\in L}p_{l}\cdot z_{s_{ 2}\delta(s_{2},t_{1})}\right)+M(1-x_{t_{1}s_{1}}) \geq \phi\cdot N\] \[a_{t_{2}s_{2}}\cdot\left(\sum_{l\in L}p_{l}\cdot z_{s_{2}\delta( s_{2},t_{2})}\right)-\phi a_{t_{2}s_{1}}\cdot\left(\sum_{l\in L}p_{l}\cdot z _{s_{1}\delta(s_{1},t_{2})}\right)+M(1-x_{t_{2}s_{2}}) \geq \phi\cdot N\.\]  
The previous constraints respectively represent TP \(t_{1}\) served by \(s_{1}\) and interfered by \(s_{2}\) and TP \(t_{2}\) served by \(s_{2}\) and interfered by \(s_{1}\).