# Achievable Rate Maximization for Multi-Relay AF Cooperative SWIPT Systems with a Nonlinear EH Model
## 2 System Model
Since \(\gamma_{R_{l},D}^{\mathrm{PS}}\) is concave and positive, \(1/\gamma_{R_{l},D}^{\mathrm{PS}}\) is convex [38]. Similarly, for the third term of the objective function in (16), \(1/\left(\gamma_{S,R_{l}}^{\mathrm{PS}}\gamma_{R_{l},D}^{\mathrm{PS}}\right)\), we first prove that \(\gamma_{S,R_{l}}^{\mathrm{PS}}\gamma_{R_{l},D}^{\mathrm{PS}}\) is concave with respect to \(\rho_{l}\) in \(\rho_{l}\in(0,1)\). From (8) and (9), \(\gamma_{S,R_{l}}^{\mathrm{PS}}\gamma_{R_{l},D}^{\mathrm{PS}}\) can be expressed as:  
\[y_{S,R_{l}}(n)=\sqrt{P_{s}}h_{S,R_{l}}x_{s}(n)+N_{R_{l},a}(n), \tag{1}\]8  
Let \(f_{2}(x)=\frac{x(1-x)}{1+x}\) and \(f_{3}(x)=\frac{x}{1+x}\). It is readily found that \({f_{2}}^{\prime\prime}(x)<0\) and \({f_{3}}^{\prime\prime}(x)<0\) for all \(x\in(0,1)\), where \({f_{2}}^{\prime\prime}(x)\) and \({f_{3}}^{\prime\prime}(x)\) are the second derivatives of \(f_{2}(x)\) and \(f_{3}(x)\), respectively. Therefore, both \(f_{2}(x)\) and \(f_{3}(x)\) are concave with respect to \(x\) in \(x\in(0,1)\). This means that \(\gamma_{S,R_{l}}^{\mathrm{PS}}\gamma_{R_{l},D}^{\mathrm{PS}}\) is the pointwise minimum of two concave functions \(C_{1,l}C_{2,l}\rho_{l}(1-\rho_{l})/(1+\rho_{l})\) and \(C_{1,l}C_{3,l}\rho_{l}/\bigl{(}1+\rho_{l}\bigr{)}\) with respect to \(\rho_{l}\), and hence, they are concave, such that \(1/\Bigl{(}\gamma_{S,R_{l}}^{\mathrm{PS}}\gamma_{R_{l},D}^{\mathrm{PS}}\Bigr{)}\) is convex. Since all \(1/\gamma_{S,R_{l}}^{\mathrm{PS}}\), \(1/\gamma_{R_{l},D}^{\mathrm{PS}}\), and \(1/\Bigl{(}\gamma_{S,R_{l}}^{\mathrm{PS}}\gamma_{R_{l},D}^{\mathrm{PS}}\Bigr{)}\) are convex, their sum is convex and problem (16) is convex. Although (16) is convex, it is non-smooth hence cannot be solved using the standard convex tools. Here, we use the proximal gradient method [39; 40] to solve (16). Let \(f_{0}(\rho_{l})=1\Big{/}\gamma_{S,R_{l}}^{\mathrm{PS}}=\bigl{(}1+1/\rho_{l} \bigr{)}/C_{1,l}\) and \(g_{0}(\rho_{l})=1\Big{/}\gamma_{R_{l},D}^{\mathrm{PS}}+1\Big{/}\Bigl{(}\gamma _{S,R_{l}}^{\mathrm{PS}}\gamma_{R_{l},D}^{\mathrm{PS}}\Bigr{)}\). Then, the problem (16) can be rewritten as:  
\[y_{S,R_{l}}(n)=\sqrt{P_{s}}h_{S,R_{l}}x_{s}(n)+N_{R_{l},a}(n), \tag{1}\]9 Figure 1:0  
Let \(h(\rho_{l})=f(\rho_{l})+g(\rho_{l})\). By using the logarithmic barrier method, the inequality constrained minimization problem (21) can be transformed into the unconstrained minimization problem given as:  
Figure 1:1  
where we use \(x\) to denote \(\rho_{l}\) for representation convenience, \(f(x)=tf_{0}(x)-\log x(1-x)\), \(g(x)=tg_{0}(x)\), where \(t>0\) is the parameter that sets the accuracy of the logarithmic barrier method [38]. The iterative algorithm based on [PERSON] and [PERSON] proximal gradient method [39; 40] for solving (22) is given in Algorithm 1 as follows, where \(\
abla f\Bigl{(}x^{(l)}\Bigr{)}\) is the gradient of \(f(x)\) at \(x^{(l)}\) and \(\mathrm{prox}_{\lambda g}(\cdot)\) is the proximal operator of the scaled function \(\lambda g\) defined by:  
Figure 1:2  
where \(\lambda>0\) is the step size that controls the extent to which the proximal operator maps points towards the minimum of \(g\)[39], \(\|\cdot\|_{2}\) is the usual Euclidean norm, \(\widehat{f}_{\lambda^{(l)}}\left(z,x^{(l)}\right)\) is given by:  
Figure 1:3  
**Remark 1**. : _Complexity and optimality of Algorithm 1. Since the proximal gradient decent satisfies \(h\Bigl{(}x^{(l)}\Bigr{)}-h(x^{*})\leq\left\|x^{(0)}-x^{*}\right\|_{2}^{2}\! \middle/2\lambda l\) for both fixed step size \(\lambda\) and backtracking step size \(\lambda^{(l)}=\beta\lambda^{(l-1)}\), the computational complexity or the convergence rate of Algorithm 1 is \(O(1/l)\) or \(O(1/e)\), which also means that an order of \(1/e\) iterations is required to obtain an \(e\)-optimalsolution for problem (22) [40; 41].