Proximal operator
In mathematical optimization, the proximal operator is an operator associated with a proper, lower semi-continuous convex function formula_1 from a Hilbert space formula_2 to formula_3, 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 formula_5 of a function enjoys several useful properties for optimization, enumerated below. Note that all of these items require formula_1 to be proper (i.e. not identically formula_7, and never take a value of formula_8), convex, and lower semi-continuous. A function is said to  be firmly non-expansive if formula_9. Fixed points of formula_10 are minimizers of formula_1: formula_12. Global convergence to a minimizer is defined as follows: If formula_13, then for any initial point formula_14, the recursion formula_15 yields convergence formula_16 as formula_17. This convergence may be weak if formula_2 is infinite dimensional. It is frequently used in optimization algorithms associated with non-differentiable optimization problems such as total variation denoising. If formula_19 is the 0-formula_20 indicator function of a nonempty, closed, convex set, then it is lower semi-continuous, proper, and convex and formula_10 is the orthogonal projector onto that set