# Multicell Interference Management in Device to Device Underlay Cellular Networks
## 5 Power Control  
In this section, we present the second stage of our approach for the power control problem in multicell D2D underlay network architectures, considering that the RBs have already been assigned to the cellular and the D2D users according to the output of the first stage of our approach. Thus, capitalizing on the outcome of the RB allocation approach, Problem (2) takes the following form:  
\[\underset{P}{\text{minimize}} \sum_{i=1}^{E}I_{i}\] (14) subject to \[0\leq P_{u}\leq P_{u}^{\max} \tag{14a}\] \[\gamma_{u}\geq\gamma_{u}^{tar} \tag{14b}\]  
All cells have already chosen their RB assignment matrices and thus Problem (14) can be decomposed to multiple optimization problems per available RB since we assume that there is no interference between different RBs due to their orthogonality. Problem (14) is equivalent to multiple interference minimization problems per RB \(k\). Then each problem for every RB \(k\) can be written as:  
\[\underset{P}{\text{minimize}} \sum_{i\in R_{k}}\sum_{\begin{subarray}{c}j\in R_{k},\\ j\
eq i\end{subarray}}G_{v}^{ijk}P_{v}+\sigma^{2}\] (15) subject to \[0\leq P_{v}\leq P_{v}^{\max} \tag{15a}\] \[\gamma_{v}\geq\gamma_{v}^{tar} \tag{15b}\]  
\(R_{k}\) is the set of cells who occupy RB \(k\) and \(v\) the user of cell \(j\) who occupies RB \(k\). There is no other user who uses the same RB at the same cell since we assume orthogonality of the available RBs. Let \(P_{v}\) be the transmission power of the the user \(v\) of cell \(j\) who uses RB \(k\).  
The objective is to minimize total interference at every RB \(k\) under the constraints of feasible transmission power and SINR requirement satisfaction of the users. Problem (15) is a typical linear programming problem. Thus, if this problem has a feasible solution this lies at the point of intersection of the Constraints (15a) and (15b). The gradient of the objective function shows the direction of the optimal change of the objective Function with respect to the independent variables of the problem i.e., transmission powers of the users sharing RB \(k\). The gradient of the objective Function (15) is positive with respect to the transmission power of every user of each cell who uses common RB \(k\). Thus, the minimum power which satisfies all Conditions (15a) and (15b) is the optimal feasible solution which minimizes the objective function of Problem (15).  
The gradient of the objective function of Problem (15) has the same sign to the total power minimization problem under the same SINR and transmission power constraints. For the problem of total power minimization, as it is already known from the literature [29], there is a distributed algorithm which converges to the optimal solution, if this solution is feasible i.e., satisfies Constraints (15a) and (15b). Thus, by applying this algorithm to Problem (15), we can find the required optimal solution.  
Let \(P_{init}\) be the initial transmission power vector of all users of the network. This is a fixed transmission power vector which could be the maximum transmission power value for each user of the network. At Line 4 of Algorithm 2, each user updates its current power in a parallel fashion. After the execution of this step, a coordination mechanism is needed in order to check the power convergence of each user.  
```
1:Counter \(t=0\), \(Converged=false\), \(P(0)=P_{init}\)
2:if\(Converged==false\)then
3:for all\(i\in R_{k}\)do
4:\(P_{u}(t+1)=\min\left\{\frac{\gamma_{i}^{\textit{true}}}{\gamma_{u}(t)P_{u}(t)},P _{u}^{\max}\right\},u\in V_{i}\), \(s_{i}(u,k)=1\)
5:endfor
6:if\(P_{u_{i}}(t+1)==P_{u_{i}}(t)\)\(\forall i\in R_{k}\)then
7:\(Converged=true\)
8:endif
9:\(t=t+1\)
10:endif
```  
**Algorithm 2**Distributed Power Control per RB \(k\)