Proximal operator

In mathematical optimization, the proximal operator is an operator associated with a proper,An (extended) real-valued function f on a Hilbert space is said to be proper if it is not identically equal to , and is not in its image. 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

showing that the proximity operator is indeed a generalisation of the projection operator.

The proximal operator is firmly non-expansive: . 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 In particular, If is differentiable then the above equation reduces to .

Notes

References

Proximal operator

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