In mathematical optimization, the proximal operator is an operator associated with a proper,[note 1] lower semi-continuous convex function
, 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.
of a proper, lower semi-continuous convex function
enjoys several useful properties for optimization.