# Proximal Operators¶

## Definition¶

Let be a proper convex function mapping the normed space to the extended real number line . The proximal operators of the functional is mapping from . It is denoted as with and defined by

The shorter notation ) is also common.

## Properties¶

Some properties which are useful to create or compose proximal operators:

Separable sum

If is separable across variables, i.e. , then

Post-composition

If with , then

Pre-composition

If with , then

Moreau decomposition

This is also know as the Moreau identity

where is the convex conjugate of .

Convec conjugate

The convex conjugate of is defined as

where denotes inner product. For more on convex conjugate and convex analysis see [Roc1970] or Wikipedia.

For more details on proximal operators including how to evaluate the proximal operator of a variety of functions see [PB2014].

## Indicator function¶

Indicator functions are typically used to incorporate constraints. The indicator function for a given set is defined as

Special indicator functions

Indicator for a box centered at origin and with width :

where denotes the maximum-norm.