In mathematical optimization, the proximal operator is an operator associated with a proper, lower semi-continuous convex function f {\displaystyle f} from a Hilbert space X {\displaystyle {\mathcal {X}}}
to [ − ∞ , + ∞ ] {\displaystyle [-\infty ,+\infty ]} , and is defined by:
prox f ( v ) = arg min x ∈ X ( f ( x ) + 1 2 ‖ x − v ‖ X 2 ) . {\displaystyle \operatorname {prox} _{f}(v)=\arg \min _{x\in {\mathcal {X}}}\left(f(x)+{\frac {1}{2}}\|x-v\|_{\mathcal {X}}^{2}\right).}
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 prox {\displaystyle {\text{prox}}} of a proper, lower semi-continuous convex function f {\displaystyle f} enjoys several useful properties for optimization.
Fixed points of prox f {\displaystyle {\text{prox}}_{f}} are minimizers of f {\displaystyle f} : { x ∈ X | prox f x = x } = arg min f {\displaystyle \{x\in {\mathcal {X}}\ |\ {\text{prox}}_{f}x=x\}=\arg \min f} . Global convergence to a minimizer is defined as follows: If arg min f ≠ ∅ {\displaystyle \arg \min f\neq \varnothing } , then for any initial point x 0 ∈ X {\displaystyle x_{0}\in {\mathcal {X}}} , the recursion ( ∀ n ∈ N ) x n + 1 = prox f x n {\displaystyle (\forall n\in \mathbb {N} )\quad x_{n+1}={\text{prox}}_{f}x_{n}} yields convergence x n → x ∈ arg min f {\displaystyle x_{n}\to x\in \arg \min f} as n → + ∞ {\displaystyle n\to +\infty } . This convergence may be weak if X {\displaystyle {\mathcal {X}}} is infinite dimensional. The proximal operator can be seen as a generalization of the projection operator. Indeed, in the specific case where f {\displaystyle f} is the 0- ∞ {\displaystyle \infty } characteristic function ι C {\displaystyle \iota _{C}} of a nonempty, closed, convex set C {\displaystyle C} we have that
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