Proximal operator
In mathematical optimization, the proximal operator is an operator associated with a proper, lower semi-continuous convex function from a Hilbert space
to, and is defined by:
For any function in this class, the minimizer of the right-hand side above is unique, hence making the proximal operator well-defined. The proximal operator is used in proximal gradient methods, which is frequently used in optimization algorithms associated with non-differentiable optimization problems such as total variation denoising.
Properties
The of a proper, lower semi-continuous convex function enjoys several useful properties for optimization.- Fixed points of are minimizers of :.
- Global convergence to a minimizer is defined as follows: If, then for any initial point, the recursion yields convergence as. This convergence may be weak if is infinite dimensional.
- The proximal operator can be seen as a generalization of the projection operator. Indeed, in the specific case where is the 0- characteristic function of a nonempty, closed, convex set we have that
- A function is firmly non-expansive if .
- The proximal operator of a function is related to the gradient of the Moreau envelope of a function by the following identity:.
- The proximity operator of is characterized by inclusion, where is the subdifferential of, given by