# ProximalOperators.jl

Proximal operators for nonsmooth optimization in Julia.
This package can be used to easily implement proximal algorithms for convex and nonconvex optimization problems such as ADMM, the alternating direction method of multipliers.

See the documentation on how to use the package.

## Installation

To install the package, use the following in the Julia command line

```
Pkg.add("ProximalOperators")
```

Remember to `Pkg.update()`

to keep the package up to date.

## Usage

With `using ProximalOperators`

the package exports the `prox`

and `prox!`

methods to evaluate the proximal mapping of several functions.

A list of available function constructors is in the documentation.

For example, you can create the L1-norm as follows.

```
julia> f = NormL1(3.5)
description : weighted L1 norm
type : Array{Complex} → Real
expression : x ↦ λ||x||_1
parameters : λ = 3.5
```

Functions created this way are, of course, callable.

```
julia> x = randn(10) # some random point
julia> f(x)
32.40700818735099
```

`prox`

evaluates the proximal operator associated with a function,
given a point and (optionally) a positive stepsize parameter,
returning the proximal point `y`

and the value of the function at `y`

:

```
julia> y, fy = prox(f, x, 0.5) # last argument is 1.0 if absent
```

`prox!`

evaluates the proximal operator *in place*,
and only returns the function value at the proximal point:

```
julia> fy = prox!(y, f, x, 0.5) # in-place equivalent to y, fy = prox(f, x, 0.5)
```

## References

N. Parikh and S. Boyd (2014), *Proximal Algorithms*,
Foundations and Trends in Optimization, vol. 1, no. 3, pp. 127-239.

S. Boyd, N. Parikh, E. Chu, B. Peleato and J. Eckstein (2011), *Distributed Optimization and Statistical Learning via the Alternating Direction Method of Multipliers*, Foundations and Trends in Machine Learning, vol. 3, no. 1, pp. 1-122.

## Credits

ProximalOperators.jl is developed by
Lorenzo Stella
and Niccolò Antonello
at KU Leuven, ESAT/Stadius,
and Mattias Fält at Lunds Universitet, Department of Automatic Control.