## 3 Problem Formulation  
### Mathematical Formulation  
In this paper, we aim to develop the optimal solution of content placement and user association that balances network throughput and the backhaul load simultaneously. Considering the fairness of UEs, the network throughput of cell edge UEs is used as the performance metric. Thus the objective function is a tradeoff between network throughput of cell edge UEs and backhaul load.  
\[\max_{\mathbf{X},\mathbf{Y}} \sum_{m\in M^{\Omega}}\log_{10}(R_{m})-\lambda\frac{\sum_{m\in M^ {\Omega}}V_{m}}{|M^{\Omega}|}\] (15) s.t. \[\text{C1:}\quad 1\leq|B_{m}^{\Omega}|\leq N,\forall m\in M^{\Omega}\] \[\text{C2:}\quad\sum_{m}x_{m,b}\leq N_{\text{RB}},\forall b\in B^{ \Omega},\forall m\in M^{\Omega}\] \[\text{C3:}\quad\sum_{m}x_{m,b}(1-\mathbf{q}_{m}\mathbf{y}_{b}^{T})U_{ \text{back}}\leq C_{b}^{\max},\forall b\in B^{\Omega}\]  
where the constraint C1 indicates that a specific UE \(m\) should be served by at least one BS, meanwhile the total number of BSs that cooperatively serve a specific UE should not be more than a given number \(N\) by considering the tradeoff between throughput gain and backhaul load due to the joint transmission of CoMP; C2 indicates that the total number of RBs allocated to UEs associating with a specific BS is limited to the maximum \(N_{\text{RB}}\); C3 indicates that aggregate backhaul load of BS \(m\) should not be over the backhaul capacity \(C_{b}^{\max}\).  
\(\lambda\geq 0\) in Equation (15) is a coefficient that influences the balance between network throughput and backhaul load. A larger \(\lambda\) suggests that we prefer improving network throughput than reducing backhaul load, and vice versa. Let \(\sum_{m\in M^{\Omega}}\log_{2}(R_{m}^{(0)})\) and \(\frac{\sum_{m\in M^{\Omega}}V_{m}^{(0)}}{|M^{\Omega}|}\) denote sum of logarithm of data rate and averaged load on backhaul links when \(\lambda=0\), respectively. We define \(\lambda\) with \(\sum_{m\in M^{\Omega}}\log_{2}(R_{m}^{(0)})\) and \(\frac{\sum_{m\in M^{\Omega}}V_{m}^{(0)}}{|M^{\Omega}|}\) as benchmarks, and then \(\lambda\) can be given by:  
\[\lambda=\frac{\sum_{m\in M^{\Omega}}\log_{2}(R_{m}^{(0)})\left|M^{\Omega} \right|\mu}{\sum_{m\in M^{\Omega}}V_{m}^{(0)}}. \tag{16}\]  
where \(\mu\in[0,1]\) is a weight factor used for adjusting \(\lambda\).  
The problem in Equation (15) is a constrained non-convex combinatorial optimization problem, which requires extraordinary high complexity to trace its optimal solution. To obtain a practical solution, we decompose the problem into two steps based on the cross-entropy (CE) method. In this paper, the CE method is chosen because it is a simple, efficient, and general method for solving a great variety of estimation and optimization problems, especially NP-hard combinatorial deterministic and stochastic problems [15]. First, we minimize the backhaul load of the system under the assumption of the conventional \(N\)-Best user association strategy (By \(N\)-Best user association strategy, a CoMP UE will associate with \(N_{\max}\) BSs which have \(N\) best SINRs [16]) and propose a content placement algorithm based on cross entropy, which is termed as the CPCE algorithm. Subsequently, under the given content placement strategy, we propose an user association algorithm based on cross entropy, which is referred to as UACE in the rest of this paper.