Proximal operator
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.