Algorithm 4.

Pseudocode for N Steps of the CP Algorithm Instance for Quadratic Reweighted Constrained TpV Minimization

1: INPUT: data g, data-error tolerance ∊, exponent p, and smoothing parameter η

2: INPUT: algorithm parameters ν; λ

3: L ← ∥(X, ν▽)∥2; τ ← 1/L; σ ← 1/L; θ ← 1; n ←  0

4: initialize f0, y0, and ${\overrightarrow{\mathbf{z}}}_0$ to zero vectors

5: ${\stackrel{‒}{\mathbf{f}}}_0\leftarrow {\mathbf{f}}_0$

6: repeat

7: ${\mathbf{y}}_n^{\prime }\leftarrow {\mathbf{y}}_n+\sigma (X{\stackrel{‒}{\mathbf{f}}}_n-\mathbf{g})$

8: ${\mathbf{y}}_{n+1}\leftarrow \max ({\Vert {\mathbf{y}}_n^{\prime }\Vert }_2-\sigma ∊,0)\frac{{\mathbf{y}}_n^{\prime }}{{\Vert {\mathbf{y}}_n^{\prime }\Vert }_2}$

9: $\mathbf{w}\leftarrow {\left(\sqrt{{\eta }^2+{\mid \nabla {\stackrel{‒}{\mathbf{f}}}_n\mid }^2}∕\eta \right)}^{p-2}$

10: ${\overrightarrow{\mathbf{z}}}_n^{\prime }={\overrightarrow{\mathbf{z}}}_n+\sigma \nu \nabla {\stackrel{‒}{\mathbf{f}}}_n$

11: ${\overrightarrow{\mathbf{z}}}_{n+1}\leftarrow {\overrightarrow{\mathbf{z}}}_n^{\prime }∕(1+\sigma {\nu }^2∕(2\mathbf{w}\lambda \left)\right)$

12: ${\mathbf{f}}_{n+1}\leftarrow {\mathbf{f}}_n-\tau (X^T{\mathbf{y}}_{n+1}+\nu {\nabla }^T{\overrightarrow{\mathbf{z}}}_{n+1})$

13: ${\stackrel{‒}{\mathbf{f}}}_{n+1}\leftarrow {\mathbf{f}}_{n+1}+\theta ({\mathbf{f}}_{n+1}-{\mathbf{f}}_n)$

14: n ← n + 1

15: until n ≥ N

16: OUTPUT: fN

17: OUTPUT: w, yN, and ${\overrightarrow{\mathbf{z}}}_N$ for evaluating cPDquad and conditions 3.