w.r.t., the constraints on the previous slide, two optimization problems can be formulated. $$\\$$ (P1) uplink sum-power minimization with objective $\sum_{j=1}^N p_j$ and (P2) uplink sum-rate maximization under QoS with objective $\sum_{j=1}^N \alpha_{ij} r_{ij}$. $$\\$$ Since the UEs are homogeneous the QoS is balanced in P1 i.e., each UE will have the same QoS after convergence. $$\\$$ $r_{th}=0$ in P2 corresponds to sum-rate maximization. Here multiuser diversity is exploited. This may result in an increase in the objective value while some UEs are dropped. $$\\$$ $$\\$$

Note : P1 and P2 are mixed integer nonlinear optimization problems in $( \alpha_{ij},p_j )$.