## 4 Optimization Analysis  
In this part, the optimization analysis is realized with regards to three users in the NOMA system and the objective function can simply reformulated as follows [24]:  
\[\max\limits_{\alpha}R_{Sum}=R_{n}+R_{m}+R_{f} \tag{15}\]  
\[S.t.\] \[(2^{R_{min}}-1)-\left|h_{k}\right|^{2}\rho\left(\alpha_{m}-(2^{R_{min}}-1 )\sum\limits_{i=1}^{m-1}\alpha_{i}\right)\leq 0\] \[\alpha_{n}+\alpha_{m}+\alpha_{f}-1\leq 0\] \[\alpha_{n},\ \alpha_{m},\alpha_{f}\ \geq 0\]  
where \(m=2,\ 3\) and \(R_{min}\) is the lowest rate required in the system. According to the analysis above, the constraints can also be represented as follows:  
\[C_{1}(\alpha)=\alpha_{n}+\alpha_{m}+\alpha_{f}-1 \tag{16}\]  
\[C_{2}(\alpha)=(2^{R_{min}}-1)-\rho\left|h_{f}\right|^{2}\left(\alpha_{f}-(2^{ R_{min}}-1)(\alpha_{n}+\alpha_{m}\right) \tag{17}\]  
\[C_{3}(\alpha)=(2^{R_{min}}-1)-\rho|h_{m}|^{2}\left(\alpha_{m}-(2^{R_{min}}-1)( \alpha_{n}\right) \tag{18}\]  
The constraints \(C_{1}(\alpha),\ C_{2}(\alpha)\) & \(C_{3}(\alpha)\) are linear in terms of \(\alpha\), then \(C_{1}(\alpha),\ C_{2}(\alpha)\) & \(C_{3}(\alpha)\) are convex. Hence, \(\
abla R_{Sum}(\alpha)\) & \(\
abla^{2}R_{Sum}(\alpha)\) need to be calculated [21]. Initially, we can find the first derivative for \(R_{Sum}(\alpha)\) in Equation (14) with respect to each of the power coefficients \(\alpha_{n}\), \(\alpha_{m}\), and \(\alpha_{f}\). After some mathematical processing, \(\
abla R_{Sum}(\alpha)\) can be represented as follows [24,25]:  
Based on the objective function, and \(\frac{\partial R_{S_{sum}}(\alpha)}{\partial\alpha_{i}}\) and \(\frac{\partial^{2}R_{S_{sum}}(\alpha)}{\partial\alpha_{i}z}\), it can be proved that the objective function is concave and has a unique global maximum [23]. The Lagrange function and the KKT necessary conditions could be employed to achieve optimal power factors [25; 26]:  
\[\mathcal{L}\Big{(}\alpha_{n},\alpha_{m},\alpha_{f},\mu_{1},\mu_{2},\mu_{3} \Big{)}=R_{sum}-\mu_{1}C_{1}(\alpha)-\mu_{2}C_{2}(\alpha)-\mu_{3}C_{3}(\alpha) \tag{27}\]  
where \(\mu_{1},\ \mu_{2},\) and \(\mu_{3}\) represent Lagrange multipliers for the three user scenarios. Optimality conditions can be written as follows:  
\[\frac{\partial R_{sum}}{\partial\alpha_{n}}-\mu_{1}\frac{\partial C_{1}( \alpha)}{\partial\alpha_{n}}-\mu_{2}\frac{\partial C_{2}(\alpha)}{\partial \alpha_{n}}-\mu_{3}\frac{\partial C_{3}(\alpha)}{\partial\alpha_{n}}=0 \tag{28}\]  
\[S.t.\]0  
\[S.t.\]1  
Slackness conditions can be represented as follows:  
\[S.t.\]2  
\[S.t.\]3  
\[S.t.\]4  
Lagrange multipliers also need to satisfy the following:  
\[S.t.\]5  
In the subsequent steps, Lagrange multipliers should be proved to be positive. This could be accomplished as follows:  
\[S.t.\]6  
\[S.t.\]7  
\[S.t.\]8  
\[S.t.\]9  
\[(2^{R_{min}}-1)-\left|h_{k}\right|^{2}\rho\left(\alpha_{m}-(2^{R_{min}}-1 )\sum\limits_{i=1}^{m-1}\alpha_{i}\right)\leq 0\]0  
Based on Equations (35)-(39), this can be substituted in the optimality conditions for Lagrange as follows:  
\[(2^{R_{min}}-1)-\left|h_{k}\right|^{2}\rho\left(\alpha_{m}-(2^{R_{min}}-1 )\sum\limits_{i=1}^{m-1}\alpha_{i}\right)\leq 0\]1  
\[(2^{R_{min}}-1)-\left|h_{k}\right|^{2}\rho\left(\alpha_{m}-(2^{R_{min}}-1 )\sum\limits_{i=1}^{m-1}\alpha_{i}\right)\leq 0\]2  
\[(2^{R_{min}}-1)-\left|h_{k}\right|^{2}\rho\left(\alpha_{m}-(2^{R_{min}}-1 )\sum\limits_{i=1}^{m-1}\alpha_{i}\right)\leq 0\]3Let \(\beta 1=\rho\Big{|}h_{f}\Big{|}^{2}(2^{R_{min}}-1)\), \(\beta 2=\rho|h_{m}|^{2}(2^{R_{min}}-1)\), \(\gamma 1=\Big{(}-\rho\Big{|}h_{f}\Big{|}^{2}\Big{)}\), and \(\gamma 2=\Big{(}-\rho|h_{m}|^{2}\Big{)}\).