## 4 Proposed Technique
### Combinatorial Optimisation Algorithm  
Furthermore, the formulated EE problem in (3a) is classified as a non-linear fractional programming, thus, solving the problem by applying general mathematical approaches greatly increases its complexity; therefore, a significantly less-complex iterative algorithm, based on [PERSON]'s method can be utilised to obtain the solution as presented in Algorithm 1. [PERSON]'s theorem is generally used to simplify the non-linear complex fractional problem into a simpler iterative algorithm [28]. First, the objective function is converted from fractional into a non-fractional function with a parameter \(\tau\) (refer to Appendix A for derivation). Initially, the parameter \(\tau\) is set to 0 as stated in step 1 of Algorithm 1. Along with it, the maximum number of iterations and convergence condition are also initialised accordingly. The optimal EE \(\tau^{*}\) is only achievable upon fulfilment of convergence condition, otherwise, the obtained EE value \(\tau\) is used in the following iteration to solve the problem as shown in steps 10-15 of Algorithm 1.  
The Lagrangian function is evaluated in each iteration as indicated in step 7 of Algorithm 1 based on the BS On/Off statuses and updated transmit power allocation values. Along with the dual decomposition method, steps 4-9 in Algorithm 1 show the combined application of sub-gradient method to obtain the optimal power allocation value. Simple characteristics, i.e., pre-defined constant step size, of sub-gradient method make it more preferable to be applied compared to the gradient method, which uses a computed instantaneous step size [29]. Once the formulated problem is divided into sub-problems with the aid of Lagrange duality, each subgradient is updated iteratively with a constant step size \(\delta(n)\) as indicated in step 8 of Algorithm 1. Consequently, each of the Lagrange multiplier values is also updated in every iteration and the convergence condition is evaluated each time. The condition is violated only if the difference between the previous and current values of the Lagrangian function drops below the convergence threshold as can be seen in step 9 of the algorithm.  
```
1:Set maximum number of iterations \(T_{out}\); convergence condition \(\epsilon\) & \(\tau^{(1)}=0\)
2:Set iteration index, \(n=1\)
3:Set the \(b_{j}\) status of each base station using one of the BS-Sw methods:
4:Optimal (Binomial theorem) * As described in Algorithm 2.
5:Heuristic 1 (Distance-based) * As described in Algorithm 3.
6:Heuristic 2 (Random) * As described in Algorithm 5.
7:for\(1\leq n\leq T_{out}\)do
8:Initialise Lagrange multipliers to an arbitrarily large positive value.
9:Update power allocation (obtained from dual decomposition), Equation (9).
10:Evaluate Lagrangian function (of the primal objective function) \(f(n)\), Equation (5).
11:Update Lagrange multipliers with small positive step size \(\delta(n)\): \(\alpha_{i}^{(n+1)}=[\alpha_{j}^{(n)}-\delta(n)*(P_{j,max}-\sum_{i}a_{i}p_{i}p_{ j})]^{+}\) \(\beta_{i}^{(n+1)}=[\beta_{i}^{(n)}-\delta(n)*(\frac{\sum_{i}a_{i}p_{i}p_{j}p_{ i}\beta_{i}}{B_{n}N_{0}}-SNR_{i,thr})]^{+}\)
12:Repeat steps 6-8 until \(|f(n+1)-f(n)|\leq\epsilon\) or maximum number of iterations is reached.
13:if\(R(p^{(n)},b^{(n)})-\tau^{(n)}p^{Total}(p^{(n)},b^{(n)})<\epsilon\)then
14:set \(\tau^{*}=\tau^{(n)}\)
15:break
16:else
17:set \(\tau^{(n+1)}=\frac{R(p^{(n)},b^{(n)})}{p^{Total}(p^{(n)},b^{(n)})}\)
18:\(n=n+1\)
19:endif
20:endfor
```  
**Algorithm 1** PABS-Sw algorithm.