# Cross-Layer Optimization for Heterogeneous MU-MIMO/OFDMA Networks
## 4 Optimization Framework
### Decision Variables
#### 4.1.2 Transmission Mode  
Thus far, we have defined the variables related to the data rate. In the following, we will discuss the variables related to MU-MIMO and OFDMA transmission modes. Since at this stage, we are assuming that nodes cannot use MU-MIMO and OFDMA at the same time, it is necessary to introduce some variables to indicate the transmission mode of the currently active link or node. To do this, we define several variables that begin with \(m\) and \(o\). Two binary variables, \(m_{llf}\) and \(o_{llf}\), are used to indicate whether flow \(f\) on link \(l\) at time slot \(t\) is invoked in MU-MIMO mode or OFDMA mode, respectively. Since the nodes cannot activate MU-MIMO and OFDMA at the same time, we have the following constraint:  
\[m_{llf}+o_{llf}=z_{llf},\hskip 28.452756 pt\forall l,t,f. \tag{12}\]  
Recall that \(z_{llf}\) is already limited to one by constraint (7). Similarly, we introduce the following variables: \(m_{it}^{x}\), \(o_{it}^{x}\), \(m_{it}^{y}\) and \(o_{it}^{y}\), each of which indicates whether node \(i\) at time slot \(t\) is a MU-MIMO transmitter, an OFDMA transmitter, a MU-MIMO receiver, or an OFDMA receiver, respectively. Then, assuming the property of half-duplex transmission, we have:  
\[m_{it}^{x}+m_{it}^{y}+o_{it}^{x}+o_{it}^{y}\leq 1,\hskip 28.452756 pt\forall i,t. \tag{13}\]  
We can readily see that these variables are defined using \(m_{llf}\) and \(o_{llf}\), as follows: \(m_{it}^{x}=\max_{l\in L_{i}^{\text{out}},f}(m_{llf})\), \(m_{it}^{y}=\max_{l\in L_{i}^{\text{in}},f}(m_{llf})\), \(o_{it}^{x}=\max_{l\in L_{i}^{\text{out}},f}(o_{llf})\), and \(o_{it}^{y}=\max_{l\in L_{i}^{\text{in}},f}(o_{llf})\). By employing the same technique used in constraint (9), we can convert them to the following linear constraints:  
\[m_{llf}\leq m_{it}^{x}\leq\sum_{l,f}m_{llf},\hskip 28.452756 pt \forall i,l\in L_{i}^{\text{out}},t,f \tag{14}\] \[m_{llf}\leq m_{it}^{y}\leq\sum_{l,f}m_{llf},\hskip 28.452756 pt \forall i,l\in L_{i}^{\text{in}},t,f\] (15) \[o_{llf}\leq o_{it}^{x}\leq\sum_{l,f}o_{llf},\hskip 28.452756 pt \forall i,l\in L_{i}^{\text{out}},t,f\] (16) \[o_{llf}\leq o_{it}^{y}\leq\sum_{l,f}o_{llf},\hskip 28.452756 pt \forall i,l\in L_{i}^{\text{in}},t,f. \tag{17}\]  
In addition, to model the behavior of the node in each subchannel, we introduce the following set of binary variables: \(m_{itk}^{x}\), \(m_{itk}^{y}\), \(o_{itk}^{x}\) and \(o_{itk}^{y}\). Each of these indicates whether node \(i\) transmits or receives data over subchannel \(k\) at time slot \(t\) using MU-MIMO or OFDMA, respectively. Similarly, we can also express them in terms of the variables defined above; for example, \(m_{itk}^{x}\) is equivalent to \(\max_{l\in L_{i}^{\text{out}},f}(z_{llf})\) && \(m_{it}^{x}\). The relevant constraints are given below:  
\[m_{itk}^{x}\leq m_{it}^{x},\hskip 28.452756 pt\forall i,t,k\] (18) \[m_{itk}^{y}\leq m_{it}^{y},\hskip 28.452756 pt\forall i,t,k\] (19) \[o_{itk}^{x}\leq o_{it}^{x},\hskip 28.452756 pt\forall i,t,k\] (20) \[o_{itk}^{y}\leq o_{it}^{y},\hskip 28.452756 pt\forall i,t,k\] (21) \[z_{llfk}+m_{it}^{x}-1\leq m_{itk}^{x}\leq\sum_{l\  
From the assumption that streams of different transmission modes cannot be processed together, we have the following constraints:  
\[m^{y}_{itk}+o^{x}_{itk}\leq 1, \forall i,j\in N^{\text{int}}_{i},t,k \tag{26}\] \[o^{y}_{itk}+m^{x}_{itk}\leq 1, \forall i,j\in N^{\text{int}}_{i},t,k \tag{27}\]  
where \(N^{\text{int}}_{i}\) is a set of nodes within the interference range of node \(i\).  
The remaining variables, \(s^{x}_{itk}\) and \(s^{y}_{itk}\), indicate the number of spatial streams allocated to node \(i\) in subchannel \(k\) at time slot \(t\), and will be discussed in the following subsection.