Inmathematical optimization, theproximal operator is anoperator associated with a proper,^{[note 1]}lower semi-continuousconvex function${\displaystyle f}$ from aHilbert space${\displaystyle \mathcal{X}}$to${\displaystyle [-\mathrm{\infty },+\mathrm{\infty }]}$, and is defined by:^[1]

${\displaystyle {\mathrm{prox}}_f(v)=\arg \underset{x\in \mathcal{X}}{min}\left(f(x)+\frac{1}{2}\Vert x-v{\Vert }_{\mathcal{X}}^2\right).}$

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 astotal variation denoising.

The${\displaystyle \text{prox}}$ of a proper, lower semi-continuous convex function${\displaystyle f}$ enjoys several useful properties for optimization.

• Fixed points of${\displaystyle {\text{prox}}_f}$ are minimizers of${\displaystyle f}$:${\displaystyle \{x\in \mathcal{X}|{\text{prox}}_{f}x=x\}=\arg minf}$.
• Global convergence to a minimizer is defined as follows: If${\displaystyle \arg minf\ne \varnothing }$, then for any initial point${\displaystyle x_0\in \mathcal{X}}$, the recursion${\displaystyle (\mathrm{\forall }n\in \mathbb{N})x_{n+1}={\text{prox}}_f{x}_n}$ yields convergence${\displaystyle x_n\to x\in \arg minf}$ as${\displaystyle n\to +\mathrm{\infty }}$. This convergence may be weak if${\displaystyle \mathcal{X}}$ is infinite dimensional.^[2]
• The proximal operator can be seen as a generalization of theprojection operator. Indeed, in the specific case where${\displaystyle f}$ is the 0-${\displaystyle \mathrm{\infty }}$ characteristic function${\displaystyle {\iota }_C}$ of a nonempty, closed, convex set${\displaystyle C}$ we have that

showing that the proximity operator is indeed a generalisation of the projection operator.

• A function isfirmly non-expansive if${\displaystyle (\mathrm{\forall }(x,y)\in {\mathcal{X}}^2)\Vert {\text{prox}}_{f}x-{\text{prox}}_{f}y{\Vert }^2\le ⟨x-y,{\text{prox}}_{f}x-{\text{prox}}_{f}y⟩}$.
• The proximal operator of a function is related to the gradient of theMoreau envelope${\displaystyle M_{\lambda f}}$ of a function${\displaystyle \lambda f}$ by the following identity:${\displaystyle \mathrm{\nabla }{M}_{\lambda f}(x)=\frac{1}{\lambda }(x-{\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{x}}_{\lambda f}(x))}$.

• The proximity operator of${\displaystyle f}$ is characterized by inclusion${\displaystyle p={\mathrm{prox}}_f(x)\iff x-p\in \mathrm{\partial }f(p)}$, where${\displaystyle \mathrm{\partial }f}$ is thesubdifferential of${\displaystyle f}$, given by

${\displaystyle \mathrm{\partial }f(x)=\{u\in {\mathbb{R}}^N\mid \mathrm{\forall }y\in {\mathbb{R}}^N,(y-x{)}^{\mathrm{T}}u+f(x)\le f(y)\}}$ In particular, If${\displaystyle f}$ is differentiable then the above equation reduces to${\displaystyle p={\mathrm{prox}}_f(x)\iff x-p=\mathrm{\nabla }f(p)}$.

• Proximal gradient method

• TheProximity Operator repository: a collection of proximity operators implemented inMatlab andPython.
• ProximalOperators.jl: aJulia package implementing proximal operators.
• ODL: a Python library forinverse problems that utilizes proximal operators.

• Mathematical optimization

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