## 3 Infeasible Target SINRs and the Best Effort Service  
When the set of target SINRs is feasible, the best response of cell \(n\) to a given intercell interference can be found to be rearranging (4) as follows [22]:\[p_{n}^{d}=\Gamma_{n}^{d}\frac{(1+\beta)I_{n}^{d}+(P_{I}+\kappa+\beta)p_{n}^{u}g_{ mn}^{du}+\sigma_{o}^{2}}{g_{mn}^{du}},\ \forall n\in\mathcal{N}, \tag{5}\]  
\[p_{n}^{u}=\Gamma_{n}^{u}\frac{(1+\beta)I_{n}^{u}+(\kappa+\beta)p_{n}^{d}g_{mn}^ {ud}+\sigma_{o}^{2}}{g_{mn}^{u}},\ \forall n\in\mathcal{N}.\]  
If all nodes simultaneously iterate the above equations, they will converge to a fixed point which achieves the set of target SINRs \(\{(\Gamma_{n}^{d},\Gamma_{n}^{u})\}_{n=1}^{N}\)[22]. However, in case of infeasibility, the iteration will diverge and thus a solution is needed for such a case. In this section, we propose a game based solution that guarantees QoS achievement in case of feasible target SINRs and provides the best effort service otherwise. In case of infeasibility, it is obviously more desirable to reach an equilibrium point that minimizes the gap between the right-hand side (RHS) and the left-hand side (LHS) of both equations in (5), in all cells (If the target SINRs are feasible, the gap would be zero in all cells).Therefore, we define \(\mathbf{f}_{n}\) as a column vector whose rows are the difference between RHS and LHS of the first and second equations in (5). Thus,  
\[\mathbf{f}_{n}=\left[ \begin{array}{c}p_{n}^{d}-\Gamma_{n}^{d}\frac{(1+\beta) I_{n}^{d}+(P_{I}+\kappa+\beta)p_{n}^{u}g_{mn}^{du}+\sigma_{o}^{2}}{g_{mn}^{du}}\\ p_{n}^{u}-\Gamma_{n}^{u}\frac{(1+\beta)I_{n}^{u}+(\kappa+\beta)p_{n}^{ud}g_{mn }^{ud}+\sigma_{o}^{2}}{g_{mn}^{du}}\end{array} \right]. \tag{6}\]  
For a given amount of interference \((I_{n}^{d},\ I_{n}^{u})\), each player \(n\in\mathcal{N}\) should be able to find the best response \((p_{n}^{d},\ p_{n}^{u})\) such that the norm of \(\mathbf{f}_{n}\) is minimized. According to the power constraints, the problem of intercell interference in full duplex wireless system can be formulated in two different ways as follows:  
**P1: Least difference with separate power constraints**  
\[\underset{p_{n}^{d},\ p_{n}^{u}}{\text{minimize}}\ \|\mathbf{f}_{n}\|^{2}\] subject to: \[0\leq p_{n}^{d}\leq p_{max}^{d},\] \[0\leq p_{n}^{u}\leq p_{max}^{u}.\]
**P2: Least difference with joint power constraints**  
\[\underset{p_{n}^{d},\ p_{n}^{u}}{\text{minimize}}\ \|\mathbf{f}_{n}\|^{2}\] subject to: \[p_{n}^{d},\ p_{n}^{u}\geq 0,\] \[p_{n}^{d}+p_{n}^{u}\leq p_{max}^{d}+p_{max}^{u}.\]  
**Theorem 1**: _The problems **P1** and **P2** are convex on \(p_{n}^{d}\) and \(p_{n}^{u}\)._  
Proof. : First, the constraint inequalities in each problem constitute polyhedron set, which is a convex set [23]. Second, the objective function can be written as a least square objective function that is \(\|\mathbf{f}\|^{2}=\|\mathbf{Ap}_{n}-\mathbf{b}\|^{2}\), where \(\mathbf{p}_{n}=[p_{n}^{d}\ p_{n}^{u}]^{T}\),  
\[\mathbf{A}=\left[ \begin{array}{cc}1&-\Gamma_{n}^{d}\frac{(P_{I}+\kappa+\beta )g_{mn}^{du}}{g_{mn}^{du}}\\ -\Gamma_{n}^{u}\frac{(\kappa+\beta)g_{mn}^{ud}}{g_{mn}^{du}}&1\end{array} \right], \tag{7}\]\[\mathbf{b}=\left[ \begin{array}{c}\Gamma_{n}^{d}\frac{(1+\beta)t_{n}^{d}+\sigma_{n }^{2}}{\frac{g_{m}^{d}}{g_{m}^{d}}}\\ \Gamma_{n}^{u}\frac{(1+\beta)t_{n}^{u}+\sigma_{n}^{2}}{g_{m}^{u}}\end{array}  \right]. \tag{8}\]  
It is worth mentioning that \(\mathbf{A}\) is a full rank matrix, and, hence, it is nonsingular. Therefore, the Hessian of the objective function (i.e., \(\frac{\partial^{2}\|\mathbf{f}\|^{2}}{\partial\mathbf{p}_{n}^{2}}=2\mathbf{A}^ {T}\mathbf{A}\)) is positive definite which means that the objective function is a strictly convex function in \(\mathbf{p}_{n}\)[23]. Thus, the convexity of the problems \(\mathbf{P1}\) and \(\mathbf{P2}\) follows directly. As a result, the above two problems can be casted as least-square optimization problems with inequality constraints [24; 25]. Now, the non-cooperative game for interference management is formally introduced.