Table 1 Immediate OPA algorithm introduction

1:

Initialization: set i=0,λ_m(0)>0,μ(0)>0,ξ_m(0)>0;

\({\alpha _{1}}\left (0 \right) > 0, {\alpha _{2}}\left (0 \right) > 0,{\alpha _{3}}\left (0 \right) > 0;{I^{th}} > 0,R_{\min }^{m} > 0\);

2:

Solve the optimization problem (13) to obtain \(p_{k}^{m,n},\forall k\),

thus we can get the throughput of the secondary system;

3:

Update the Lagrange multiplier by (19) ∼ (21);

4:

Update the transmission power \(p_{k}^{m,n}\) by (18);

5:

Go to 2 until \(\left | {\hat p_{k}^{m,n}\left ({i + 1} \right) - \hat p_{k}^{m,n}\left (i \right)} \right | \le \varepsilon \), where ε represents

iteration precision usually a very small positive constant;

6:

End: The optimal transmission power \(p_{k}^{m,{n^ * }}\) can be calculated

by (18), and take \(p_{k}^{m,{n^ * }}\) into (11), the optimal throughput can

be obtained.