Proximal operator

In mathematical optimization, the proximal operator is an operator associated with a proper, lower-semicontinuous convex function from a Hilbert space to , and is defined by: [1]

For any function in this class, the minimizer of the righthandside above is unique, hence making the proximal operator well-defined. The prox of a function enjoys several useful properties for optimization, enumerated below. Note that all of these items require to be proper (i.e. not identically , and never take a value of ), convex, and lower-semicontinuous.
  • Firmly nonexpansive:
  • Fixed points of are minimizers of : .
  • Global convergence to a minimizer: If , then for any initial point , the recursion yields convergence as . This convergence may be weak if is infinite dimensional.[2]

It is frequently used in optimization algorithms associated with non-differentiable optimization problems such as total variation denoising.

If is the 0- indicator function of a nonempty, closed, convex set, then it is lower-semicontinuous, proper, and convex and is the orthogonal projector onto that set.

See also


  1. Neal Parikh and Stephen Boyd (2013). "Proximal Algorithms" (PDF). Foundations and Trends in Optimization. 1 (3): 123–231. Retrieved 2019-01-29.
  2. 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.
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