Algorithm 1 SCA with randomization

Initialize $ϵ,n=1$ Solve (12) to obtain ${\mathbf{w}}_{\mathrm{ap}}$               ▹ Performing SDR method Initialize ${\tilde{\mathbf{w}}}^{\left(0\right)}={\mathbf{w}}_{\mathrm{ap}}$ for$l=1,\cdots ,L$do                  ▹ Performing randomization   Generate ${\mathbf{w}}_l={\mathbf{w}}_{\mathrm{ap}}+{\xi }_l$, ${\mathbf{w}}_l={\mathbf{w}}_l/\parallel {\mathbf{w}}_l\parallel$   if ${min}_k{\parallel {\mathbf{w}}_l^H{\mathbf{H}}_k\parallel }^2\ge {min}_k\parallel {\tilde{\mathbf{w}}}^{\left(0\right)H}{\mathbf{H}}_k{\parallel }^2$ then     Update ${\tilde{\mathbf{w}}}^{\left(0\right)}\leftarrow {\mathbf{w}}_l$   end if end for Set the initial solution ${\tilde{\mathbf{w}}}^{\left(1\right)}$ by solving (21) with ${\tilde{\mathbf{w}}}^{\left(0\right)}$ while$\parallel {\tilde{\mathbf{w}}}^{\left(n\right)}-{\tilde{\mathbf{w}}}^{(n-1)}\parallel \ge ϵ$do   Solve (21) to obtain $\tilde{\mathbf{w}}$ using ${\tilde{\mathbf{w}}}^{\left(n\right)}$        ▹ Performing SCA algorithm   Update ${\tilde{\mathbf{w}}}^{(n+1)}\leftarrow \tilde{\mathbf{w}}$   Update $n\leftarrow n+1$ end while return$\tilde{\mathbf{w}}$ as a solution of the algorithm