# Cooperative Admission Control with Network Coding in 5G Underlying D2D-Satellite Communication
## Problem Formulation
### Joint Admission Control-ATSPSN
\(\bigcup_{*}\) include set of constraints from C1 to C13,\(P_{*}=(z_{k},R,P_{d}^{l},P_{c}^{l},P_{s}^{l})\) and \(\gamma=R\cup\
u^{l}\). Qualification of constraints cannot be violated by fixing values of \(\gamma\) and \(P_{*}\) is non-strictly convex for fixed \(\gamma\). The primitive problem is given as:  
\[ \begin{split} min(P_{*})-&\Delta_{t}\left(\gamma^{ j},P_{*}\right)\\ &\Delta_{t}|\Delta_{*}\left(\gamma^{j},P_{*}\right)\leqslant 0\end{split}  \tag{13}\]  
By solving (13), we get \(P_{*}\) for master problem. The primitive problem provides a lower bound while the master problem gives an upper bound. The master problem consists of two steps to achieve linearization. In the first step, we project on integer space \(\Delta_{*}\). Mathematically, represented as:  
\[min_{(\gamma^{j})}min_{(P_{*})}\Delta_{t}\left(\gamma^{j},P_{*}\right)\leqslant 0 \tag{14}\]  
We can Formulate (14) further as  
\[ \begin{split} min_{(\gamma^{j})}-\eta(\gamma^{j})\\ \eta(\gamma^{j})=min_{(P_{*})}-\Delta_{t}(\gamma^{j},P_{*})\\ \text{subject to}\\ \Delta_{*}\left(\gamma^{j},P_{*}\right)\leqslant 0\end{split}  \tag{15}\]  
Further, we introduce a variable as \(e\) and projection of solution (15) in \(j\)th iteration will become as  
\[min_{(\gamma^{j},P_{*},e)}. \tag{16}\]  
subject to \[e\geq -\Delta_{t}\left(\gamma^{j},P*\right)-\bigtriangledown\Delta_{t}\left( \gamma^{j},P*\right)\left( \begin{array}{c}P-P_{*}\\ \gamma-\gamma^{j}\end{array} \right)\] such that \[-\Delta_{*}\left(\gamma^{j},P*\right)-\bigtriangledown\Delta_{t}\left( \gamma^{j},P*\right)\left( \begin{array}{c}P-P_{*}\\ \gamma-\gamma^{j}\end{array} \right)\leq 0.\]  
Equation (16) generate lower limits, leads to MILP. The same formulation is further solved by GAMS branch-cut and NOMAD to find an optimal solution. The Algorithm 1 give brief steps for the ATSPSN GAMS branch cut. The details about ATSPSN NOMAD and branch bound along with problem formulation are discussed in the conference version of this paper [25]. ```
1. \(j\gets 1\)
2. Initializes \(\gamma^{j}\), \(\Delta_{t}\) and constraint space \(S^{j}\)
3. \(e\gets 10^{-3}\)
4. Convergence \(\leftarrow\) Violated
5. while Convergence==Violated then for non violated space
6. \(P_{*}\leftarrow \begin{cases}\arg\min\ \ -\Delta_{t}\left(\gamma^{j},P_{*}\right) \\ \text{subject to}\ \ \ \Delta_{*}\left(\gamma^{j},P_{*}\right)\leq 0\\ \end{cases} \)
7. Upper limit \(\leftarrow\Delta_{t}\left(\gamma^{j},P_{*}\right)\)
8. \(\left(\gamma^{*},P^{*},e*\right)\leftarrow \begin{cases}\arg\min\ \ \left\{e \right\}_{\gamma,P,e}\\ \text{subject to}\\ e\geq-\Delta_{t}\ \left(\gamma_{j},P_{*}\right)\\ -\bigtriangledown\Delta_{t}\left(\gamma^{j},P_{*}\right)\left(\begin{array}[] {c}P-P_{*}\\ 0\end{array}\right)\\ \Delta_{t}\left(\gamma^{j},P_{*}\right)\\ -\bigtriangledown\Delta_{t}\left(\gamma^{j}P_{*}\right)\left(\begin{array}[] {c}P-P_{*}\\ 0\end{array}\right)\leq\ 0\\ \Delta_{t_{SAT}}^{j}\left(\gamma_{j},P_{*}\right)\geq\Delta_{t_{D2D}}^{j} \left(\gamma_{j},P_{*}\right)\end{cases} \)
9. Lower limit \(\gets e\)
10. if \(\left\langle upperlimit-lowerlimit\right\rangle\leq\epsilon\) then
11. Convergence holds
12. else
13. \(j\gets j+1\)
14. \(\gamma^{j}\leftarrow\gamma^{*}\)
15. \(S\leftarrow\) Violated region
16. end if
17. end while
```  
**Algorithm 1**ATSPSN (GAMS-Branch-cut)