# Multicell Interference Management in Device to Device Underlay Cellular Networks
## 3 Problem Formulation  
In the following, we formulate the problem of interference minimization in the joint space of RBs and uplink transmission power.  
\[\underset{S,P}{\text{minimize}} \sum_{i=1}^{E}I_{i}\] (2) subject to \[\sum_{k=1}^{K}s_{i}(u,k)=1 \tag{2a}\] \[0\leq P_{u}\leq P_{u}^{\max}\] (2b) \[\gamma_{u}\geq\gamma_{u}^{tar}\] (2c) \[\sum_{u\in V_{i}}s_{i}(u,k)=1,\quad\forall k\in K,\forall i\in E \tag{2d}\]  
where  
\[I_{i}=\sum_{u\in V_{i}}\sum_{k=1}^{K}s_{i}(u,k)\sum_{j\
eq i}\sum_{v\in V_{j}} s_{j}(v,k)\cdot P_{v}\cdot G_{v}^{i,k}+\sigma^{2} \tag{3}\]  
is the total sensed interference from all users at cell \(i\), caused by any other user of the other cells of the system who share common RBs with the users of cell \(i\) and  
\[s_{i}(u,k)=\begin{cases}1&\text{if user $u$ of cell $i$ uses RB $k$}\\ 0&\text{otherwise}\end{cases} \tag{4}\]  
User \(u\) belongs to cell \(i\) and user \(v\) belongs to cell \(j\). \(P_{u}\) denotes the transmission power of user \(u\) of the cell \(i\) at the chosen RB from user \(u\). Since every user is assumed to use exactly one RB we can omit the exponent \(k\) from \(P_{u}^{k}\) without disambiguation. \(P_{u}^{max}\) is the maximum transmission power of user \(u\). \(\gamma_{u}\) is the SINR at the receiver of user \(u\). Similarly the exponent \(k\) is omitted from \(\gamma_{u}^{k}\). \(\gamma_{u}^{tar}\) is a positive real number denoting the SINR requirement of user \(u\).  
\(S=\{s_{i}\}_{i=1,\ldots,|E|}\) and \(P=\{P_{u}\}_{u\in V_{i},i=1,\ldots,|E|}\), where \(|E|\) is the number of the cells in the network.  
The first Constraint (2a) concerns the fact that each user of each cell (either physical or virtual) chooses exactly one RB. Constraints (2b) and (2c) concern the acceptable range of values for the transmission power and SINR levels. The final Constraint (2d) is related to the orthogonality requirement between users of the same cell. It is stressed here that based on the consideration of the additional virtual cells, a D2D user belongs to the corresponding cell (which essentially is a single user cell) and not to the physical cell that is placed within. In a nutshell, in order to solve the aforementioned optimization problem we search for feasible combinations of RB and transmission power assignments which minimize the total interference in the network.  
The above problem is a mixed integer programming problem since some of the decision variables are integer numbers and the rest of them i.e. transmission powers, continuous variables. This problem is a combinatorial optimization problem and its complexity is known to be NP-hard [31]. For this reason, we aim at efficient approximation algorithms to solve the problem. The proposed approach is to solve the problem in two stages. In the first stage, we consider that all users have chosen a fixed transmission power and we search for the optimal RB assignment. We propose a game theoretic approach which solves the problem iteratively. At each iteration every player of the game makes a decision regarding its RB which maximizes its utility function while simultaneously minimizes the total interference in the network. After a finite number of iterations the algorithm converges to the desirable NE point of the game. At the second stage, the problem of total interference minimization can be decomposed to multiple interference minimization problems per RB. Certain number of users share a RB and create interference one to another. Our goal is to determine users' optimal transmissionpower allocation towards minimizing the total interference per RB while satisfying the minimum SINR requirements of the respective users.