Table 2 Algorithm for solving problem (13)

1:

Initialize λ≽0,μ≥0 such that \(\mu \mathbf {I} - \sum \nolimits _{j\in \mathcal {U}_{E}} \lambda _{j} \hat {\mathbf {H}}_{ji}^{H}\hat {\mathbf {H}}_{ji} \succ 0, \forall i\)

2:

Repeat

3:

Compute \(\tilde {\mathbf {S}}_{i}(\boldsymbol \lambda,\mu) \forall i\) using (14)

4:

Compute subgradient of g(λ,μ):

5:

\([\mathbf {t}]_{m} = Q_{N+m} - \sum \nolimits _{i\in \mathcal {U}_{I}}\text {Tr}(\hat {\mathbf {H}}_{(N+m)i}\tilde {\mathbf {S}}_{i}\hat {\mathbf {H}}_{(N+m)i}^{H})\) for 1≤m≤M

6:

\([\mathbf {t}]_{M+1} = \text {Tr}(\tilde {\mathbf {S}}_{i}) - (P_{\max } - P^{tx}_{c})\)

7:

Update λ,μ using the ellipsoid method [41] subject to the following:

λ≽0,μ≥0 and \(\mu \mathbf {I} - \sum \nolimits _{j\in \mathcal {U}_{E}} \lambda _{j} \hat {\mathbf {H}}_{ji}^{H}\hat {\mathbf {H}}_{ji} \succ 0, \forall i\)

8:

Until dual variables converge