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.

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.

Here is a list of the available functions.

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

References

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

2. 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.