# Efficient Resource Allocation for Beam-Hopping-Based Multi-Satellite Communication Systems
## 3 Offline Resource Allocation Scheme
### Power Resource Allocation  
After allocating the temporal and frequency resource, all the matrices \(\mathbf{A}_{m}^{j}\) in problem (5) are determined. Observing the form of the objective function in this problem, as both the numerator and denominator of the SINR \(\gamma_{m,nk}^{j}\) contain optimization variable \(p_{m,nk}^{j}\), it is evident that this problem is still non-convex. We use the well-known tool of SCA to attain a suboptimal solution [29]. First, the negative value of sum throughput can be written as  
\[-\sum\limits_{j=1}^{J}\sum\limits_{m=1}^{M}R_{m}^{j}=\sum\limits_{j,m,k,n}B_{sc} \Big{[}U_{m,nk}^{j}(\mathbf{p})-V_{m,nk}^{j}(\mathbf{p})\Big{]} \tag{8}\]  
where \(U_{m,nk}^{j}(\mathbf{p})\) and \(V_{m,nk}^{j}(\mathbf{p})\) are convex functions given by:  
\[U_{m,nk}^{j}(\mathbf{p})=\log_{2}\left(\sum\limits_{(i,q)\
eq(j,m)}a_{q,nk}^{i }p_{q,nk}^{i}H_{jm,k}^{i}+N_{0}\right), \tag{9}\]  
\[V_{m,nk}^{j}(\mathbf{p})=\log_{2}\left(\sum\limits_{(i,q)}a_{q,nk}^{i}p_{q,nk} ^{i}H_{jm,k}^{i}+N_{0}\right). \tag{10}\]  
Now when we give an approximation point \(\overline{\mathbf{p}}\), a convex minimization problem can be formulated as  
\[\min\limits_{\mathbf{P}}\sum\limits_{j,m,k,n}B_{sc}\Big{[}\widetilde{U}_{m,nk }^{j}(\mathbf{p},\overline{\mathbf{p}})-V_{m,nk}^{j}(\mathbf{p})\Big{]}, \tag{11}\]  
where \(\widetilde{U}_{m,nk}^{j}(\mathbf{p},\overline{\mathbf{p}})\) can be expressed as  
\[\widetilde{U}_{m,nk}^{j}(\mathbf{p},\overline{\mathbf{p}})=U_{m,nk}^{j}( \overline{\mathbf{p}})+\bigtriangledown U_{m,nk}^{j}(\overline{\mathbf{p}})( \mathbf{p}-\overline{\mathbf{p}})^{T}. \tag{12}\]  
In the above formula, \(\bigtriangledown U_{m,nk}^{j}(\overline{\mathbf{p}})\) is a derivative vector, and its elements are given by  
\[\frac{\partial U_{m,nk}^{j}(\overline{\mathbf{p}})}{p_{q,nk}^{i}}=\frac{a_{q, nk}^{i}H_{jm,k}^{i}}{\ln 2\left(\sum\limits_{(i,q)\
eq(j,m)}a_{q,nk}^{i} \overline{p}_{q,nk}^{i}H_{jm,k}^{i}+N_{0}\right)}. \tag{13}\]  
Note that the constraint function \(R1\) in problem (5) is still non-convex, and we can convert it into a convex function in the same way  
\[T_{m}^{j}-\sum\limits_{j,m,k,n}B_{sc}\Big{[}\widetilde{U}_{m,nk}^{j}(\mathbf{ p},\overline{\mathbf{p}})-V_{m,nk}^{j}(\mathbf{p})\Big{]}\leq 0. \tag{14}\]  
Finally, the original non-convex problem is transformed into the following convex problem:  
\[ \begin{split}&\min\limits_{\mathbf{P}}\sum\limits_{j,m,k,n}B_{sc} \Big{[}\widetilde{U}_{m,nk}^{j}(\mathbf{p},\overline{\mathbf{p}})-V_{m,nk}^{ j}(\mathbf{p})\Big{]}\\ & s.t.\ R1:T_{m}^{j}-\sum\limits_{j,m,k,n}B_{sc}\Big{[}\widetilde{ U}_{m,nk}^{j}(\mathbf{p},\overline{\mathbf{p}})-V_{m,nk}^{j}(\mathbf{p})\Big{]} \leq 0,\forall j,\forall m,\\ & R2:\sum\limits_{m=1}^{M}\sum\limits_{k=1}^{K}\sum\limits_{n=1} ^{N}p_{m,nk}^{j}-P_{max}\leq 0,\forall j,\\ & R3:p_{m,nk}^{j}\geq 0,\forall j,\forall m,\forall n,\forall k, \end{split}  \tag{15}\]  
We use the classical Lagrange dual method to solve the convex problem. First, the Lagrange function of \(\mathbf{P}\) can be expressed as  
\[ \begin{split}& L(\mathbf{P},\lambda,\mu)=\sum\limits_{j,m,k,n}B_{ sc}\Big{[}\widetilde{U}_{m,nk}^{j}(\mathbf{p},\overline{\mathbf{p}})-V_{m,nk}^{j}( \mathbf{p})\Big{]}+\\ &\sum\limits_{j=1}^{I}\sum\limits_{m=1}^{M}\lambda_{j,m}\Bigg{(} T_{m}^{j}-\sum\limits_{j,m,k,n}B_{sc}\Big{[}\widetilde{U}_{m,nk}^{j}(\mathbf{p}, \overline{\mathbf{p}})-V_{m,nk}^{j}(\mathbf{p})\Big{]}\Bigg{)}+\sum\limits_{j= 1}^{J}\mu_{j}\Bigg{(}\sum\limits_{m=1}^{M}\sum\limits_{k=1}^{K}\sum\limits_{n= 1}^{N}p_{m,nk}^{j}-P_{max}\Bigg{)}\end{split}  \tag{16}\]where \(\lambda_{j,m}\) and \(\mu_{j}\) represent the Lagrange multiplier.