## 3 Resource Allocation Algorithm  
### Convex Analysis of Resource Allocation Problem  
The signal strength strategy (SSS) is a classical association method for the terminals and APs [16]. Under the SSS association method, a terminal is associated with the closest AP and the multi-cell networks can be equally decomposed into multiple single-cell networks that have interfering signals from neighboring cells. Therefore, if the resource allocation performance of each cell is optimized, the resource allocation performance of the whole network is optimized. The problem of maximizing the throughput under resource constraints for a certain interfering cell is formulated as problem **P1**:  
\[\begin{array}{ll}\textbf{P1}:&\max&f(\textbf{x})=  
where \(f_{0}(\mathbf{x})=-f(\mathbf{x})\). The first partial derivative of \(f_{0}(\mathbf{x})\) with respect to \(x_{i}\) is expressed as:  
\[\frac{\partial f_{0}}{\partial x_{i}}=-\alpha_{i}\log_{2}\Bigg{(}\frac{1+\frac{ S_{i}}{I_{i}+\beta_{i}x_{i}}}{e^{\frac{\beta_{i}S_{i}x_{i}}{(I_{i}+\beta_{i}+ \beta_{i}x_{i})(I_{i}+\beta_{i}x_{i})}}}\Bigg{)} \tag{8}\]  
If we have the notation \(g(x_{i})=\frac{1+\frac{S_{i}}{I_{i}+\beta_{i}x_{i}}}{e^{\frac{\beta_{i}S_{i}x_ {i}}{(I_{i}+\beta_{i}+\beta_{i}x_{i})(I_{i}+\beta_{i}x_{i})}}}\), the second partial derivative of \(f_{0}(\mathbf{x})\) with respect to \(x_{i}\) is expressed as:  
\[\frac{\partial^{2}f_{0}}{\partial x_{i}^{2}}=-\frac{\alpha_{i}}{\ln 2}. \frac{1}{g(x_{i})}.\frac{\partial g(x_{i})}{\partial x_{i}} \tag{9}\]  
The second mixed derivative of \(f_{0}(\mathbf{x})\) is expressed as:  
\[\frac{\partial^{2}f_{0}}{\partial x_{i}\partial x_{j}}=\frac{\partial\Big{(} \frac{\partial f_{0}}{\partial x_{i}}\Big{)}}{\partial x_{j}}=0 \tag{10}\]  
As the first partial derivative of \(g(x_{i})\) with respect to \(x_{i}\) is expressed as:  
\[\frac{\partial g(x_{i})}{\partial x_{i}}=\frac{-S_{i}\beta_{i}(2I_{i}^{2}+2I_ {i}\beta_{i}x_{i}+2S_{i}I_{i}+S_{i}\beta_{i}x_{i})}{(I_{i}+S_{i}+\beta_{i}x_{ i})(I_{i}+\beta_{i}x_{i})^{3}e^{\frac{\beta_{i}S_{i}x_{i}}{(I_{i}+\beta_{i}+ \beta_{i}x_{i})(I_{i}+\beta_{i}x_{i})}}}<0 \tag{11}\]  
the maximum and minimum of \(g(x_{i})\) is expressed as:  
\[\begin{split} g(x_{i})_{\max}=g(x_{i})|_{x_{i}=0}=1+\frac{S_{i}} {I_{i}}>0\\ g(x_{i})_{\min}=g(x_{i})|_{x_{i}=1}=\frac{1+\frac{S_{i}}{I_{i}+ \beta_{i}}}{e^{\frac{\beta_{i}S_{i}}{(I_{i}+\beta_{i})(I_{i}+\beta_{i})}}}>0 \end{split} \tag{12}\]  
According to Equations (11) and (12), we concluded that the second partial derivative of \(f_{0}(\mathbf{x})\) with respect to \(x_{i}\) in Equation (9) is non-negative. Meanwhile, according to Equations (9) and (10), the Hessian matrix of \(f_{0}(\mathbf{x})\) is a positive semi-definite matrix or positive definite matrix. Therefore, the objective function of problem **P2** is a convex function [29]. In addition, because the inequality constraint functions of problem **P2** are affine functions, the feasibility of problem **P2** is related to a convex set [29]. In summary, problem **P2** is a convex optimization problem and the local optimal solution is the global optimal solution [29]. In theory, the global optimal solution can be surely obtained [29].