Proximal operator
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In mathematical optimization, the proximal operator is an operator associated with a proper,[note 1] lower semi-continuous convex function from a Hilbert space to , and is defined by: [1]
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
[edit]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.[2]
- The proximal operator can be seen as a generalization of the projection operator. Indeed, in the specific case where is the 0- indicator function of a nonempty, closed, convex set we have that
- showing that the proximity operator is indeed a generalisation of the projection operator.
- 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
- In particular, If is differentiable then the above equation reduces to .
Notes
[edit]- ^ 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.
References
[edit]- ^ Neal Parikh and Stephen Boyd (2013). "Proximal Algorithms" (PDF). Foundations and Trends in Optimization. 1 (3): 123–231. Retrieved 2019-01-29.
- ^ Bauschke, Heinz H.; Combettes, Patrick L. (2017). Convex Analysis and Monotone Operator Theory in Hilbert Spaces. CMS Books in Mathematics. New York: Springer. doi:10.1007/978-3-319-48311-5. ISBN 978-3-319-48310-8.
See also
[edit]External links
[edit]- The Proximity Operator repository: a collection of proximity operators implemented in Matlab and Python.
- ProximalOperators.jl: a Julia package implementing proximal operators.
- ODL: a Python library for inverse problems that utilizes proximal operators.