Table V.

Proposed Stochastic OS-SQS Algorithms with Momentum (OS-mom3). ξ_k is a Realization of a Random Variable S_k.

1: Initialize x(0) = v(0) = z(0), t0 ∈ (0, 1] and compute D.

2: for n = 0, 1, …

3: for m = 0, 1, …, M − 1

4: k = nM + m

5: Choose ${\mathbf{\Gamma }}^{(k)}\triangleq \text{diag}\left\{{\gamma }_j^{(k)}\right\}\mathrm{s}.\mathrm{t}.\{\begin{array}{ll}{\mathbf{\Gamma }}^{(0)}\succ \mathit{D}\hfill & k=0\hfill \\ {\mathbf{\Gamma }}^{(k)}⪰{\mathbf{\Gamma }}^{(k-1)},\hfill & k>0\hfill \end{array}$

6: Choose $t_{k+1}\mathrm{s}.\mathrm{t}.t_{k+1}^2{\mathbf{\Gamma }}^{(k+1)}⪯\left({\sum }_{l=0}^{k+1}{t}_l\right){\mathbf{\Gamma }}^{(k)}$

7: ${\mathit{x}}^{(\frac{k+1}{M})}=\mathcal{P}\chi [{\mathit{z}}^{(\frac{k}{M})}-{[{\mathbf{\Gamma }}^{(k)}]}^{-1}M\nabla {\mathrm{\Psi }}_{{\xi }_k}({\mathit{z}}^{(\frac{k}{M})})]$

8: ${\mathit{\upsilon }}^{(\frac{k+1}{M})}=\mathcal{P}\chi [{\mathit{z}}^{(0)}-{[{\mathbf{\Gamma }}^{(k)}]}^{-1}{\sum }_{l=0}^k{t}_{l}M\nabla {\mathrm{\Psi }}_{{\xi }_l}(\mathit{z}(\frac{l}{M}))]$

9: ${\mathit{z}}^{(\frac{k+1}{M})}={\mathit{x}}^{(\frac{k+1}{M})}+\frac{t_{k+1}}{{\sum }_{l=0}^{k+1}{t}_l}({\mathit{\upsilon }}^{(\frac{k+1}{M})}-{\mathit{x}}^{(\frac{k+1}{M})})$