## 3 Energy Efficient Joint Power Control and User Association Optimization Algorithm
### Optimal EE Iterative Algorithm Based on [PERSON]'s Theorem  
Considering load optimization of each BS and the simplification of the problem, Problem P0 can be firstly transformed into Problem P1 as follows.  
\[P1: \max\limits_{\left\{\mathbf{x},\mathbf{p},\mathbf{k}\right\}} \eta\left(\mathbf{x},\mathbf{p},\mathbf{k}\right) \tag{9}\] \[s.t. C1,C2,C3\] (10) \[C4: \sum\limits_{k}x_{jk}=k_{j},\forall j\] (11) \[C5: \sum\limits_{j}k_{j}=K \tag{12}\]  
Here, \(k_{j}\) represents the load of the BS \(j\) in the network, that is, the number of users associated with the BS \(j\). The vector \(\mathbf{k}=\left\{k_{1},k_{2},...,k_{j}\right\}\) denotes the load of the network. According to Lemma 1, it is obvious that Problem P1 can be transformed into Problem P2, with the objective function transforming from a fractional expression into integral one.  
\[P2: \max\limits_{\left\{x,p,k\right\}}T\left(\omega\right)=\sum \limits_{j}\sum\limits_{k}x_{jk}\ln\left(r_{jk}\right)-\omega P_{tot} \tag{13}\] \[s.t.C1-C5 \tag{14}\]  
Here, \(\omega\) denotes the weight coefficient and the range of the weight coefficient is \(0\leq\omega\leq\sum\limits_{k}\max\limits_{j}\left(r_{jk}\right)\Bigg{/}P_{ tot}\left(p_{j}=p_{j,\max}\right)\). The weight coefficient can denote the degree of preference for energy consumption in the objective function of Problem P2.  
**Lemma 1**.: _If and only if there is an optimal parameter \(\omega^{*}\) for Problem P2, such that \(T\left(\omega^{*}\right)=0\) holds and Problem P1 obtains the optimal \(EE\) value \(\omega^{*}\)._  
Based on the [PERSON]'s theorem [37], Lemma 1 is proved in Appendix A. In fact, Lemma 1 can be easily considered as the specific example of [PERSON]'s theorem in this paper. Problem P1 is the fractional optimization problem where the functional is given by the ratio of two integrals. With [PERSON]'s approach, the fractional optimization problem is transformed into an equivalent parametric family \(\left(\mathbf{x},\mathbf{p}\right)\) of the optimization problem, where the ratio disappears and the functional is given by the weighted difference of the numerator and the denominator of the ratio. Finally, the optimization iteration algorithm can be designed as follows to solve Problem P1.  
```
0: the maximum iteration number \(iter_{max}\), the maximum tolerance error \(\sigma>0\);
0: the optimal solution \(\mathbf{x}^{*},\mathbf{p}^{*},\mathbf{k}^{*},\omega^{*}\);  Initialize: the weight coefficient of the first iteration \(\omega\left(0\right)=\omega_{0}\), iteration number \(i=0\), the variable vector \(\mathbf{x}_{0},\mathbf{p}_{0},\mathbf{k}_{0}\) while\(T\left(\omega\left(i\right)\right)=\sum\limits_{j}x_{jk}\ln\left(r_{jk}\right)- \omega\left(i\right)P_{tot}\geq\sigma\) and \(i\leq iter_{max}\)do Update\(i=i+\mathbf{\hat{1}}\); Calculate\(\left(x\left(i\right),p\left(i\right),k\left(i\right)\right)\), by solving Problem P2 (see Algorithm 2); Update\(\omega\left(i\right)=R_{tot}\left(\mathbf{x}\left(i\right),\mathbf{p}\left(i\right), \mathbf{k}\left(i\right)\right)\)/\(P_{tot}\left(\mathbf{x}\left(i\right),\mathbf{p}\left(i\right),\mathbf{k}\left(i \right)\right)\); endwhile Return\(\omega^{*}=\omega\left(i\right)\) and \(\left(\mathbf{x}^{*},\mathbf{p}^{*},\mathbf{k}^{*}\right)=\left(\mathbf{x} \left(i\right),\mathbf{p}\left(i\right),\mathbf{k}\left(i\right)\right)\);
```  
**Algorithm 1** Optimal EE iterative algorithm