# In-Band Full-Duplex Relaying for SWIPT-Enabled Cognitive Radio Networks
## 3 Proposed Solution for SRM Problem
### Proposed Iterative Algorithm  
In this subsection, we focus on convexifying Problem (17) by applying the ICA method to (17a) and (17d). To do this, we first introduce two approximate functions as follows. * Consider the function \(f(x,y)\triangleq\frac{1}{y}\ln(1+\frac{1}{x}),\ (x,y)\in\mathbb{R}_{++}^{2}\). It can be seen that \(f(x,y)\) is convex, since its Hessian is a positive definite matrix. According to [6], we can obtain an upper bound of \(f(x,y)\) around the point \((x^{(\kappa)},y^{(\kappa)})\) as: \[f(x,y) \geq\mathtt{A}(x^{(\kappa)},y^{(\kappa)})+\mathtt{B}(x^{(\kappa)},y^{(\kappa)})x+\mathtt{C}(x^{(\kappa)},y^{(\kappa)})y\] \[:=\tilde{f}^{(\kappa)}(x,y),\] (18) where: \[\mathtt{A}(x^{(\kappa)},y^{(\kappa)}) \triangleq\frac{2}{y^{(\kappa)}}\ln\Bigl{(}1+\frac{1}{x^{(\kappa )}}\Bigr{)}+\frac{1}{\bigl{(}x^{(\kappa)}+1\bigr{)}y^{(\kappa)}},\] \[\mathtt{B}(x^{(\kappa)},y^{(\kappa)}) \triangleq-\frac{1}{x^{(\kappa)}\bigl{(}x^{(\kappa)}+1\bigr{)}y^{ (\kappa)}},\] \[\mathtt{C}(x^{(\kappa)},y^{(\kappa)}) \triangleq-\frac{1}{(y^{(\kappa)})^{2}}\ln\Bigl{(}1+\frac{1}{x^{( \kappa)}}\Bigr{)}.\]
* Similarly, an upper bound of a convex function \(g(x,y)\triangleq\frac{x^{2}}{y},\ (x,y)\in\mathbb{R}_{++}^{2}\) around the point \((x^{(\kappa)},y^{(\kappa)})\) can be found as: \[g(x,y)\geq\frac{2x^{(\kappa)}}{y^{(\kappa)}}x-\frac{(x^{(\kappa)})^{2}}{(y^{ (\kappa)})^{2}}y:=\tilde{g}^{(\kappa)}(x,y).\] (19)  
Inner approximation of (17a): We introduce a new variable \(\delta\) that satisfies the following convex constraint:  
\[\frac{1}{\delta}\leq\beta,\ \delta\geq 1. \tag{20}\]  
By substituting (20) into (16) and applying (18), the concave minorants of \(R_{\mathbf{a}}\) and \(R_{\mathbf{p}}\) at the iteration \(\kappa+1\) are respectively given as:  
\[\bar{R}_{\mathbf{a}} \geq f(t_{\mathbf{s}},\delta)\geq\tilde{f}^{(\kappa)}(t_{\mathbf{ s}},\delta):=\bar{R}_{s}, \tag{21a}\] \[\bar{R}_{\mathbf{p}} \geq R_{\mathbf{a}}+f(t_{\mathbf{p}},\delta)\geq R_{\mathbf{a}}+ \tilde{f}^{(\kappa)}(t_{\mathbf{p}},\delta):=\bar{R}_{p}. \tag{21b}\]  
It is observed that (17a) can be iteratively replaced by \(\bar{R}_{s}+\bar{R}_{p}\), which is a concave objective function. We notice that when \(\kappa\rightarrow\infty\), the inequalities in (16), (20), and (21) hold with equalities. To convexify (15), the following theorem is derived using the ICA framework. **Theorem 1**. : _For two arbitrary vectors \(\mathbf{x}\) and \(\mathbf{y}\), let \(\lambda(\mathbf{x},\mathbf{y})\triangleq\eta P_{\mathbf{p}}\|\mathbf{x}\|^{2}\| \mathbf{y}\|^{2}\). The constraints in (15a) are convexified as:_  
\[\alpha\lambda(\mathbf{h}_{p},\sqrt{\rho}\mathbf{a}_{s}^{H} \mathbf{H}_{0}\mathbf{W})+\beta\psi_{\mathtt{ST},\mathtt{R}} \leq\
u_{\mathtt{ST},\mathtt{R}}, \tag{22a}\] \[:=\tilde{f}^{(\kappa)}(x,y),\]0 (22b) \[:=\tilde{f}^{(\kappa)}(x,y),\]1 (22c) \[:=\tilde{f}^{(\kappa)}(x,y),\]2  
_where \(\
u_{\mathtt{ST},\mathtt{R}}\) and \(\
u_{\mathtt{R},\mathtt{SR}}\) are newly introduced variables and \(\psi_{\mathtt{ST},\mathtt{R}}\triangleq P_{\mathbf{p}}\left|\mathbf{a}_{s}^{H }\mathbf{h}_{p}\right|^{2}+\left\|\mathbf{a}_{s}\right\|^{2}\sigma^{2}\). Similarly, the constraints in (15b) are approximated by the following convex constraints:_  
\[:=\tilde{f}^{(\kappa)}(x,y),\]3 \[:=\tilde{f}^{(\kappa)}(x,y),\]4 (23b) \[:=\tilde{f}^{(\kappa)}(x,y),\]5 (23c) \[:=\tilde{f}^{(\kappa)}(x,y),\]6  
_where \(\
u_{\mathtt{PT},\mathtt{R}}\) and \(\
u_{\mathtt{R},\mathtt{PR}}\) are new variables and \(\phi_{\mathtt{PT},\mathtt{R}}\triangleq\lambda(f_{ps},\mathbf{a}_{p}^{H} \mathbf{h}_{s})+\lambda(\mathbf{h}_{p},\sqrt{\rho}\mathbf{a}_{p}^{H}\mathbf{H}_ {0}\mathbf{W})\)._  
Proof. : Please see Appendix A. For convenience, we define \(\mathbf{v}\triangleq\left[\
u_{\mathtt{ST},\mathtt{R}};\ \
u_{\mathtt{R},\mathtt{SR}};\ \
u_{\mathtt{PT}, \mathtt{R}};\ \
u_{\mathtt{R},\mathtt{PR}}\right]\).