The proximal operator for the sorted L1 norm, the penalty used in
SLOPE, is defined as \[
\operatorname{prox}_J (v) =
\operatorname*{arg\,min}_x\left(
J(x; \lambda) + \frac 1 2 \lVert x - v \rVert_2^2
\right)
\] where \(J(x; \lambda) = \sum_{j=1}^p
\lambda_j |\beta_{(j)}|\) is the sorted L1 norm, for which \[|\beta_{(1)}| \geq |\beta_{(2)} \geq \cdots \geq
|\beta_{(p)}.\]

## Algorithms

There are several methods for solving this proximal operator and here
we provide some benchmarks of these methods. Note that these results are
almost entirely of academic nature and are added here only to serve as
reference for others who are interested in working with SLOPE and
particularly those who might be interested in improving the performance
of these algorithms.

First, we load the packages we need,.

```
library(SLOPE)
library(tidyr)
library(dplyr)
library(bench)
```

Then we setup and run our benchmarks, letting `p`

be the
length of the vector used in the operator.

```
res <- expand_grid(
p = seq(10, 10000, length.out = 10),
i = 1:20,
method = c("stack", "pava"),
time = NA
)
set.seed(2254)
for (i in seq_len(nrow(res))) {
p <- res$p[i]
x <- rnorm(p)
lambda <- sort(runif(p), decreasing = TRUE)
time <- bench_time(sortedL1Prox(x, lambda, res$method[i]))
res$time[i] <- time[["real"]] * 1e3 # milliseconds
}
```

Finally, we summarize the results in the figure below.

```
library(ggplot2)
library(scales)
ggplot(res, aes(p, time, fill = method, col = method)) +
stat_summary(
geom = "ribbon",
fun.data = mean_se,
alpha = 0.2,
col = "transparent"
) +
stat_summary(geom = "line", fun = mean) +
scale_y_log10() +
labs(y = "Time (milliseconds)", x = expression(p))
```

As we can see, the stack-based algorithm appears to perform much
better than the PAVA one does.