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SCIENCE CHINA Information Sciences, Volume 64 , Issue 8 : 182306(2021) https://doi.org/10.1007/s11432-020-3083-5

Joint optimization of user association and resource allocation in cache-enabled terrestrial-satelliteintegrating network

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  • ReceivedMar 18, 2020
  • AcceptedOct 12, 2020
  • PublishedJun 2, 2021

Abstract


Acknowledgment

This work was supported in part by National Natural Science Foundation of China (Grant Nos. 61725103, 61701363, 61931005, U19B2025), Young Elite Scientists Sponsorship Program by CAST, and Fundamental Research Funds for the Central Universities.


Supplement

Appendix

Let $f_{0}$ denote the original problem (eq:optimization~problem) and $f_{i}\left(\forall~i\in\mathcal{M}\right)$ denote terrestrial-satellite coupling constraint (eq:backhaul~limited). The Lagrangian can be rewritten as \begin{align}\mathcal{L}\left({{\boldsymbol X},{\boldsymbol B},{\boldsymbol \lambda}}\right)=f_{0}\left({\boldsymbol X},{\boldsymbol B}\right)+\sum\limits_{i=1}^{N_{\textrm{SBS}}}\lambda_{i}f_{i}\left({\boldsymbol X},{\boldsymbol B}\right), \tag{1} \end{align} where $f_{0}\left({\boldsymbol~X},{\boldsymbol~B}\right)=\sum\nolimits_{m=1}^{N_{\textrm{SBS}}}R_{m}+\mu\sum\nolimits_{m=1}^{N_{\textrm{SBS}}}\sum\nolimits_{j=1}^{N_{\textrm{user}}}\sum\nolimits_{k=1}^{N_{\textrm{sub}}}x_{m,j,k}$ and $f_{i}\left({\boldsymbol~X},{\boldsymbol~B}\right)=C_{i}-{\sum}_{j}x_{m,j,k}(1-{{\boldsymbol~q}_{j}{\boldsymbol~y}_{i}^{\rm~T}})U_{\textrm{back}}$. Assuming $\{~{\hat{\boldsymbol~X},\hat{\boldsymbol~B}}\}~$ as any feasible point of the original problem, we guarantee the following constraint: \begin{align}\sum\limits_{i=1}^{N_{\textrm{SBS}}}\lambda_{i}f_{i}\left({\hat{\boldsymbol X},\hat{\boldsymbol B}}\right)\geq0, \forall\lambda_{i}\geq0. \tag{2} \end{align} Therefore, we have \begin{align}g\left(\boldsymbol{\lambda}\right)=\underset{\left\{ {{\boldsymbol X},{\boldsymbol B}}\right\} \in\mathcal{D}}{\sup}\mathcal{L}\left({\hat{\boldsymbol X},\hat{\boldsymbol B},\lambda}\right)=f_{0}\left({\hat{\boldsymbol X},\hat{\boldsymbol B}}\right)+\sum\limits_{i=1}^{N_{\textrm{SBS}}}\lambda_{i}f_{i}\left({\hat{\boldsymbol X},\hat{\boldsymbol B}}\right)\geq f_{0}\left({\hat{\boldsymbol X},\hat{\boldsymbol B}}\right). \tag{3} \end{align} Obviously, Eq. (3) holds for any feasible point and the optimal point is no exception [1]. Accordingly, the dual optimum is the upper bound that is as close as possible to the optimal solution, i.e., \begin{align}g^{*}\geq f_{0}^{*}, \tag{4} \end{align} where $g^{*}$ is the dual optimum and $f_{0}^{*}$ is the optimal solution of (eq:optimization~problem).


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