Algorithm 1 The optimization of the phase shifts of the STEs based on MO

Input:  ${\overline{\mathit{g}}}_n,\overline{\mathit{h}},{\tilde{\mathit{g}}}_n,\tilde{\mathit{h}},\mathit{a},{\tilde{\mathit{\phi }}}_n^{ini}$

1: Initialize $\overline{\mathit{\varphi }}$ with random phases, $i=0$

2: repeat

3:        Select the step size $\mu$

4:        Update $\mathrm{v}\mathrm{e}\mathrm{c}\left(\overline{\mathit{\varphi }}\right)$ according to (23)

5:        Update the Riemannian gradient ${\mathbf{g}}_i=\nabla f\left(\overline{\mathit{\varphi }}\right)$ according to (20)(21)

6:        Calculate ${\mathit{g}}_i^{+},{\mathit{d}}_i^{+}$ from ${\mathit{x}}_i$ to ${\mathit{x}}_{i+1}$

7:        Select Polak–Ribiere parameter ${\eta }_{i+1}$

8:        Calculate the conjugate direction ${\mathit{d}}_{i+1}=-{\mathit{g}}_{i+1}+{\eta }_{i+1}{\mathit{d}}_i^{+}$

9:            Update $i\leftarrow i+1$

10: until a stopping condition is satisfied

Output:   $\overline{\mathit{\varphi }}$