Table 1 Iterative optimization algorithm

1. Set \( {N}_{\max}^l \), \( {N}_{\max}^q \) (maximum number of iterations) and initialize q ⁽¹⁾ and q ⁽⁰⁾ _, the iteration index l = 1, k = 1, and ε ₀ > 0 (convergence tolerance).

2. While |q ^(k) − q ^{(k − 1)}| ≤ ε ₀ and \( k\le {N}_{\max}^q \) do.

3. Initialize the BS power P _B: \( {\widehat{P}}_{\mathrm{B}}^{(0)}={P}_{\mathrm{B}}^{(0)} \), initialize \( {F}_q^{(1)} \) and \( {F}_q^{(0)} \), while \( \left|{F}_q^l-{F}_q^{\left(l-1\right)}\right|\le {\varepsilon}_0 \) and \( l\le {N}_{\max}^l \)

a) Optimize over \( {P}_{{\mathrm{R}}_k} \) for a fixed P _B in Eq. (26) → \( {P}_{{\mathrm{R}}_k}^{\hat{\mkern6mu} \left(l+1\right)} \),

b) Optimize over P _B for a fixed \( {P}_{{\mathrm{R}}_k} \) in Eq. (22)→ \( {P}_{\mathrm{B}}^{\hat{\mkern6mu} \left(l+1\right)} \),

c) Calculate \( {F}_q^{\left(l+1\right)} \) and update \( {F}_q^{(l)} \),

d) Update the iteration index l = l + 1.

End while

Calculate \( {\eta}_{\mathrm{EE}}^{*(k)} \) and update q by Eq. (20), namely, q ^(k) ← q ^{(k − 1)}, update the iteration index k = k + 1.

4. End while

5. Output the current \( {\eta}_{\mathrm{EE}}^{*} \).