Table 2 Proposed RPA algorithm

1:Initialize maximum iteration number T_max; Set: iteration t=0, M>0, and L>0; Lagrangian

multipliers \(\lambda (0) > 0,\{\mu _{i}^{k}\}(0) > 0,\) and \(\{\xi _{i}^{k}\}(0) > 0\); step sizes α>0,β>0, and θ>0;

upper bound of estimation error in MU-to-FBS link is ω_i∈[0,0.003];

upper bound of estimation error in femtocell link is ε_i∈[0,0.1];

upper bound of estimation error in FU-to-MBS link is δ_i∈[0,0.003].

2:Set maximum transmit power \(p_{i}^{k,max} >0\) and initialize power \(p_{i}^{k} >0\) with different initialization

values among different FUs and MUs.

3:Define interference I_th and minimum rate \(\gamma _{i}^{k,min}\), randomly generate \(\bar G_{i}^{k}, \bar h_{ij}\) and \(\bar g_{ij}\).

4:repeat

5: for t=1 to T_maxdo

6: for m=1 to Mdo

7: for l=1 to Ldo

8: Calculate transmit power \(p_{i}^{k*}\) according to (26);

9: Calculate actual SINR received at FU-Rx according to (2) and (26);

10: Calculate I_ac according to (1) and (26);

11: Calculate Lagrange multipliers \(\lambda,\mu _{i}^{k},\) and \(\xi _{i}^{k}\) from (31) - (33);

12: end for

13: end for

13 t=t+1;

14:end for

15:until t=T_max or transmit power convergence.