proxs

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.

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