## 4 Optimization Framework  
The main aims in this part include the following: (1) present the objective function and the constraints in a standard form, (2) find a general expression for the 1 st and 2 nd derivative of the objective function, (3) based on the mathematical analysis and the derived formulas, we can inspect that \(\frac{\partial^{2}R_{sum}}{\partial\eta_{i}^{2}}\) is a negative function, which validates that the objective function is a concave with distinctive global maximum, and (4) finally, we deduce the optimal power factors for each user based on applying the Lagrange function and the KKT necessary conditions.  
On the basis of the objective function in (16) and the constraints in (17) & (19) and the fact that there are two antennas at the BS and one antenna at each user terminal, the standard optimization problem can be generally reformulated as follows [24; 25]:  
\[\max\limits_{\eta}R_{sum}=\sum\limits_{i=1}^{M}\log_{2}\left(\frac{|\mathbf{h}_{i1 }+\mathbf{h}_{i2}|^{2}P_{t}\sum\limits_{j=1}^{i-1}\eta_{j}+\sigma^{2}+|\mathbf{h}_{i1 }+\mathbf{h}_{i2}|^{2}P_{t}\eta_{i}}{|\mathbf{h}_{i1}+\mathbf{h}_{i2}|^{2}P_{t}\sum\limits _{j=1}^{i-1}\eta_{j}+\sigma^{2}}\right) \tag{20}\]  
such that  
\[\sum\limits_{j=1}^{M}\eta_{j}\leq 1\]  
\[(2^{R_{min}}-1)-\rho|\mathbf{h}_{i1}+\mathbf{h}_{i2}|^{2}\left(\eta_{i}-(2^{R_{min}}- 1)\sum\limits_{j=1}^{i-1}\eta_{j}\right)\leq 0\]  
\[\eta_{i}\geq 0\quad\forall i=1,\,2,\ldots,M\]  
In this part, the power optimisation framework is accomplished with regards to three user devices in the MISO-NOMA cell, therefore, the examined constraints can be represented as shown [25; 26]:  
\[\psi_{1}(\eta)=\eta_{n}+\eta_{m}+\eta_{f}-1 \tag{21}\]\[\psi_{2}(\eta)=(2^{R_{min}}-1)-\rho\Big{|}\mathbf{h}_{f1}+\mathbf{h}_{f2}\Big{|}^{2}\Big{(} \eta_{f}-(2^{R_{min}}-1)(\eta_{m}+\eta_{n})\Big{)} \tag{22}\]  
\[\psi_{3}(\eta)=(2^{R_{min}}-1)-\rho|\mathbf{h}_{m1}+\mathbf{h}_{m2}|^{2}\Big{(}\eta_{m}-( 2^{R_{min}}-1)(\eta_{n})\Big{)} \tag{23}\]  
Since the constraints \(\psi_{1}(\eta),\ \psi_{2}(\eta)\ \&\ \psi_{3}(\eta)\) are linear in terms of \(\eta\), they are considered convex.  
Typically, to prove that the objective function \(R_{Sum}\) is concave with a distinctive global maximum, we need to find the first derivative \(\frac{\partial R_{Sum}}{\partial\eta_{i}}\) and the second derivative \(\frac{\partial^{2}R_{Sum}}{\partial\eta_{i}2}\) of the objective function [3; 24]. The first derivative of the objective function can be deuced in general form as follows [23]:  
\[\begin{array}{rl  
Given the fact that \(\left|\mathbf{h}_{n}\right|^{2}>\left|\mathbf{h}_{m}\right|^{2}>\left|\mathbf{h}_{f}\right|^{2}\), we can demonstrate that the analyzed constraints are feasible [3] and after a few mathematical manipulations the closed form for the power factors \(\eta_{f}\), \(\eta_{m}\), and \(\eta_{n}\) can be deduced as follows [27]:  
\[\eta_{f}=\left(\frac{(2^{R_{min}}-1)}{2^{R_{min}}}\right)\left(1+\frac{1}{ \rho\left|\mathbf{h}_{f1}+\mathbf{h}_{f2}\right|^{2}}\right) \tag{30}\]  
\[\eta_{m}=\left(\left(\frac{(2^{R_{min}}-1)}{2^{R_{min}}}\right)\left(1+\frac{1 }{\rho\left|\mathbf{h}_{m1}+\mathbf{h}_{m2}\right|^{2}}\right)-\left(\frac{2^{R_{min} }-1}{2^{R_{min}}}\right)^{2}\left(1+\frac{1}{\rho\left|\mathbf{h}_{f1}+\mathbf{h}_{f2 }\right|^{2}}\right)\right) \tag{31}\]  
\[\eta_{n}=1-\left(\eta_{m}+\eta_{f}\right) \tag{32}\]