Algorithm 1 Alternating Optimization Stochastic Beamforming Algorithm

1: function AO-SBA ($(\mathit{b}$, ${\left({\beta }_m\right)}_{m=1}^M$, S, Z, S, τ, Nm, Ter, eps,  $\mathsf{\omega }, \mu )$ 2: $m \leftarrow 0,{\mathit{x}}_{Sc}={[0,\dots ,0]}^{\mathrm{T} },\mathrm{w}\leftarrow {[{\mathbf{1}}^{\mathrm{T}},{\mathbf{0}}^{\mathrm{T}}]}^{\mathrm{T}}$ 3: while $m<M$ do 4: $m$ ← $m$ + 1, $n$ ← 0, $k$ ← 0 5: ${ err}_x$ ← $1/eps$, ${err}_w$ ← $1/eps$ 6:     while $n<N_m$ and ${err}_x\ge T_{er}$ do 7:        $n$ ← $n$ + 1, ${\mathit{x}}_{old}$ ← ${\mathit{x}}_{Sc}$ 8:       Create: ${\mathsf{\delta }}_x$ ∼ $B(1, 1/2)$ with values ±1 9:       ${\mathit{x}}_{Sc}^{+}={\mathit{x}}_{Sc}+{\beta }_m{\mathit{\delta }}_x$, ${\mathit{x}}_{Sc}^{-}={\mathit{x}}_{Sc}-{\beta }_m{\mathit{\delta }}_x$ 10:     ${\mathit{L}}_{\mathit{x}}^{+}=\mathit{L}\left({(\mathit{x}}_{Sc}^{+} ,\mathit{w})\right), {\mathit{L}}_{\mathit{x}}^{-}=\mathit{L}\left({(\mathit{x}}_{Sc}^{-} , \mathit{w})\right)$ 11:     ${\mathit{\xi }}_x={\displaystyle \frac{1}{2{\beta }_m}} \left({\mathit{L}}_{\mathit{x}}^{+}-{\mathit{L}}_{\mathit{x}}^{-}\right) 1\oslash {\mathit{\delta }}_x$ 12:     ${\mathit{x}}_{Sc}={\mathit{x}}_{Sc}-\tau {\mathit{\xi }}_x, {err}_x={\displaystyle \frac{{‖{\mathit{x}}_{Sc}-{\mathit{x}}_{old}‖}_2}{{‖{\mathit{x}}_{old}‖}_2+eps}}$ 13: end while 14: while $k<N_m$ and ${err}_v\ge T_{er}$ do 15:       $k$ ← $k$ + 1, ${\mathit{w}}_{old}$ ← $\mathit{w}$ 16:       Create: ${\mathit{\delta }}_w$ ∼ $B(1, 1/2)$ with values ±1 17:       ${\mathit{w}}^{+}=\mathit{w}+{\beta }_m{\mathit{\delta }}_w, {\mathit{w}}^{-}=\mathit{w}-{\beta }_m{\mathsf{\delta }}_w$ 18:       ${\mathit{L}}_{\mathit{v}}^{+}=\mathit{L}\left({\mathit{x}}_{Sc} ,{\mathit{w}}^{+}\right), {\mathit{L}}_{\mathit{x}}^{-}=\mathit{L}\left({\mathit{x}}_{Sc} ,{\mathit{w}}^{-}\right)$ 19:       ${\mathit{\xi }}_w={\displaystyle \frac{1}{2{\beta }_m}} \left({\mathit{L}}_{\mathit{v}}^{+}-{\mathit{L}}_{\mathit{v}}^{-}\right)1\oslash {\mathit{\delta }}_w$ 20:       $\mathit{w}=\mathit{w}-\tau {\mathit{\xi }}_w, {err}_w={\displaystyle \frac{{‖\mathit{w}-{\mathit{w}}_{old}‖}_2}{{‖{\mathit{w}}_{old}‖}_2+eps}}$ 21: end while 22: while $real \left({\mathit{Z}}_{in}^j\right)>0,$ do Given by (3.3) and (3.4) 23:   $p$ ← $p$ + 1, $jϵ\left\{1, 2, 3, \dots ,N\right\}$ 24: Go to function AO-SBA 25:      while 0 < $\eta$ < 1 26:                $p$ ← $p$ + 1 27:                $\mathsf{\omega }+\mu =1,$ 28:                $\eta =1-{{\mathsf{\Gamma }}_T}^2$, ${\mathsf{\Gamma }}_T=\sqrt{1-{\displaystyle \frac{P_{rad}}{P_{in}}}}$ 29:      end while 30: end while 31:   $Er\left(m\right)$← $L\left({\mathit{x}}_{Sc} ,\mathit{w}\right)$ Given by (3.18) and (3.19) 32: end while 33: end function Ensure: ${\mathit{E}\mathit{r}}_{1:M}, {\mathit{x}}_{Sc}, \mathit{w},\eta$