## 4 Iterative Algorithm using Parametric Optimization  
Another mathematically less cumbersome technique that does not require the transformation is an iterative solution technique of the fractional programming problem based on the Dinkelbach method [39]. In this algorithm, the objective of the fractional programming problem is first transformed into a parametric optimization problem, then \(\varepsilon\)-optimal solutions are obtained iteratively, where \(\varepsilon\) is the tolerance of convergence. Let \(q\in\mathbb{R}\) denote a real-valued parameter such that \(F(\boldsymbol{p},q)=\max_{p\in S}\left[f(p)-qg(p)\right]\). Therefore, \(F(\boldsymbol{p},q)\) is continuous for \(q\in\mathbb{R}\), strictly monotonic and concave over \(\mathbb{R}\)[39]. There is a one-to-one relationship between the solutions of the fractional programming formulation and iterative concave programming with parametric objective as stated by the following theorem:  
**Dinkelbach Theorem**[39]**:  
\(q^{*}=\frac{f(p^{*})}{g(p^{*})}=\max_{p\in\ S}\frac{f(p)}{g(p)}\) if and only if, \(\max_{p\in\ S}f(p)-q^{*}g(p)=f(p^{*})-q^{*}g(p^{*})=0\). Therefore, the problem formulated in Equation (3) can be solved by replacing the fractional objective function with the parametric objective function \(F(\mathbf{p},q)\) using \(\varepsilon\)-optimal Dinkelbach algorithm as given in Algorithm 1. ``` Initialization:  Parameter \(q=0\), tolerance \(\varepsilon=10^{-6}\), iteration \(i=0\), maximum iteration \(I=100\), Convergence = false, Define:\(F(\mathbf{p},q)=\sum_{j=1}^{N}C_{j}-q\left(p_{c}+\sum_{j=1}^{N}p_{j}\right)\) while (Convergence = false) and \((i\leq I)\)do \(\mathbf{p}\leftarrow\arg\max_{\mathbf{p}}\{F(\mathbf{p},q)\mid C_{1}\) to \(C_{4}\) of Equation \((\ref{eq\(\gamma_{j}^{ss}=-10\) dB, number of PU channels, \(M=10\), the rate threshold, \(C^{th}=5\times 10^{6}\) bits/sec, interference threshold, \(I_{i}^{th}=10^{-3}\), interference probability threshold \(I_{i}^{P}=0.9\), bandwidth of each PU, \(B=1\) MHz, interference contribution of each PU is Rayleigh distributed with standard deviation \(10^{-4}\). For a given set of data and channel realization, all the constraints must satisfy together and they form a region where all the possible solution points lie in. However, for a particular set of channel samples, there may not be any feasible region at all. Meaning, there is no solution point that can simultaneously satisfy all the constraints. For that channel scenario and constraint thresholds, the SU cannot access the channel. One or more thresholds must be relaxed in order to transmit for that channel conditions. The channel access probability is defined as the average number of samples for which a feasible region and hence an optimal solution exists for the problem. First let us see the how the three main constraints, _i.e._, the interference, rate and power affect the feasible region and consequently the optimal solution of the problem. Figures 2 and 3 show two typical scenarios for two different rate thresholds respectively, where we show the feasible regions and optimal solution points. Note that as mentioned previously the feasible region is the set of solution points where all the constraints are simultaneously satisfied. However, one of the solution point in the feasible region will optimize the objective function and it is the optimal energy-efficiency. In these figures, the common area inside hatched lines is the feasible region and the solid circular point is the optimal solution point (optimal energy efficiency point), where the value of energy efficiency is the maximum. These plots show how the rate threshold \(C^{th}\) affects the feasible region and the optimal energy-efficiency. The values for different parameters for Figures 2 and 3 are as follows: number of subcarriers, \(N=2\), number of PU, \(M=1\), bandwidth of each subcarrier, \(\Delta f=1\) MHz, signal-to-noise and interference ratio (SINR) for subcarriers are \(1.3\) and \(0.62\).