# Power-Efficient Resource Allocation for Active STAR-RIS-Aided SWIPT Communication Systems
## 3 Optimization of Algorithm Design
### Passive Beamforming Optimization  
Secondly, we consider the optimization of \(Q_{k},k\in\{t,r\}\) given \(\{w_{i},v_{j}\}\) and define \(c_{t,i}=\text{diag}(f_{t,i}^{H})Gw_{i}\), \(c_{r,j}=\text{diag}(f_{r,j}^{H})Gv_{j}\). We also define the vector of transmission and reflection coefficients as \(q_{k}=[P_{1}\sqrt{\beta_{k}^{k}}e^{i\beta_{k}^{k}},P_{2}\sqrt{\beta_{2}^{k}}e^ {i\beta_{2}^{k}},...,P_{M}\sqrt{\beta_{M}^{k}}e^{i\beta_{M}^{k}}]^{H},\forall k \in\{t,r\}\). Let \(C_{t,i}=c_{t,i}c_{t,i^{\prime}}^{H},C_{r,j}=c_{r,j}c_{r,j}^{H},Q_{k}=q_{k}(q_{ k})^{H},\forall k\in\{t,r\}\) meet \(Q_{k}\succeq 0\), \(diag(Q_{k})=\beta^{k}\), \(rank(Q_{k})=1\), where \(\beta^{k}\triangleq[P_{1}^{2}\beta_{1}^{k},P_{2}^{2}\beta_{2}^{k},...,P_{M}^{ 2}\beta_{M}^{k}]\), \(\forall k\in\{t,r\}\).  
The problem is next solved using a penalty-based algorithm. We transform problem (5) into the following optimization problem:  
\[\min_{Q_{k},\beta_{k}}\eta\sum_{k\in\{t,r\}}(||Q_{k}||_{*}-||Q_{k}||_{2}) \tag{18}\]  
\[\begin{array}{l}s.t.\ \overline{\
abla}_{i}\sum\limits_{j\
eq i,j\in K_{t }}\text{TR}(C_{t,i}Q_{t})-\text{TR}(C_{t,i}Q_{t})+||f_{t,i}^{H}Q_{t}||^{2} \overline{\
abla}_{i}\sigma_{RIS}^{2}\\ +\overline{\
abla}_{i}\sigma_{i}^{2}\leq 0,\forall i\in K_{t}\end{array} \tag{19}\]  
\[\sum_{k\in K_{E}}\text{TR}(C_{r,j}Q_{r})+\sum_{k\in K_{l}}\text{TR}(C_{r,j}Q_ {r})\geq\overline{E}_{j},\forall j\in K_{E} \tag{20}\]  
\[diag(Q_{k})=\beta^{k}\,,\forall k\in\{t,r\} \tag{21}\]  
\[Q_{k}\succeq 0,\forall k\in\{t,r\} \tag{22}\]  
\[0\leq\beta_{m}^{t},\beta_{m}^{r}\leq\beta_{\max}\,\beta_{m}^{t}+\beta_{m}^{r} \leq\beta_{\max},\forall m\in\overline{M} \tag{23}\]  
where the nonconvex rank-one constraint \(rank(Q_{k})=1\) is equivalently written as an equational constraint:  
\[||Q_{k}||_{*}-||Q_{k}||_{2}=0,\forall k\in\{t,r\} \tag{24}\]  
We next relax the equation constraint (24) to a penalty item attached to the target function. In Equation (24), there are \(||Q_{k}||_{*}=\sum_{i}\sigma_{i}(Q_{k})\) and \(||Q_{k}||_{2}=\sigma_{i}(Q_{k})\) standing for the nuclear and spectral paradigms, respectively, and \(\sigma_{i}(Q_{k})\) is the \(i\)-th greatest singular value of array \(Q_{k}\). For arbitrary Hermitian matrices \(Q_{k}\in\overline{H}^{M},Q_{k}\succeq 0\), there will always be \(||Q_{k}||_{*}-||Q_{k}||_{2}\geq 0\). The equation holds true only when the rank of \(Q_{k}\) is 1.  
When the \(Q_{k}\) rank is not one, \(\eta>0\) is the penalty factor of target function. A small value is first used to initialize \(\eta\) to find a feasible point, after which the iterative process gradually increases \(\eta\) to a value enormous enough to ultimately obtain a practicable rank-one solution. The algorithm ends when the following situations occur:  
\[\max\{||Q_{k}||_{*}-||Q_{k}||_{2},\forall k\in\{t,r\}\}\leq\varepsilon_{1} \tag{25}\]  
where \(\varepsilon_{1}\) denotes the accuracy that satisfies the predefined constraints of the equation. For any penalty factor \(\eta>0\), problem (18) remains nonconvex due to the nonconvexity of the target function. It is converted to (26) using the SCA method below.  
For a given point \(Q_{k}^{(n)}\), a convex upper boundary for the penalty item is gained using a first-order Taylor expansion during the \(n\)-th iteration process:  
\[\Big{|}\Big{|}Q_{k}\Big{|}\Big{|}_{*}-\Big{|}\Big{|}Q_{k}\Big{|}_{2}\leq\Big{|} \Big{|}Q_{k}\Big{|}\Big{|}_{*}-\overline{Q}_{k}^{(n)} \tag{26}\]  
where \(\overline{Q}_{k}^{(n)}\triangleq\Big{|}\Big{|}Q_{k}^{(n)}\Big{|}\Big{|}_{2}+ \text{TR}[\overline{u}(Q_{k}^{(n)})(\overline{u}(Q_{k}^{(n)}))^{H}(Q_{k}-Q_{k} ^{(n)})]\), \(\overline{u}(Q_{k}^{(n)})\) is the eigenvector corresponding to the largest eigenvalue of \(Q_{k}^{(n)}\).