Algorithm 3 FastProxSL1

input: Nonnegative and nonincreasing sequences y and λ.

while y − λ is not nonincreasing do

Identify nondecreasing and nonconstant subsequences, that is, segments i : j such that $\begin{array}{lll}{y}_i-{\mathrm{\lambda }}_i\le y_{i+1}-{\mathrm{\lambda }}_{i+1}\le \cdots \le y_j-{\mathrm{\lambda }}_j\hfill & \text{and}\hfill & y_i-{\mathrm{\lambda }}_i<y_j-{\mathrm{\lambda }}_j.\hfill \end{array}$ (2.4)   Replace the values of y and λ over such segments by their average value: for k ∈ {i, i + 1, …, j} $\begin{array}{ll}{y}_k\leftarrow \frac{1}{j-i+1}{\displaystyle \sum _{i\le k\le j}{y}_k,}\hfill & {\mathrm{\lambda }}_k\leftarrow \frac{1}{j-i+1}{\displaystyle \sum _{i\le k\le j}{\mathrm{\lambda }}_k.}\hfill \end{array}$

end while

output: x = (y − λ)+.