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  "body": "\n## Definition\n\nLet $H$ be a real [[Hilbert space|HM2HHE3J51]] with norm $\\norm{\\cdot}$.\nLet $f \\colon H \\to (-\\infty, +\\infty]$ be a [[convex|07Q7CNM5WF]], [[lower semicontinuous|RN4G8SGP6P]] function that is not identically $+\\infty$.\nThe *proximal operator* of $f$ is the map $\\operatorname{prox}_f \\colon H \\to H$ given by\n$$\\operatorname{prox}_f(v) = \\operatorname*{arg\\,min}_{x \\in H} \\left( f(x) + \\tfrac12 \\norm{x - v}^2 \\right).$$\n\n## Remarks\n\nThe minimizer exists and is unique, since the function being minimized is lower semicontinuous, strongly convex and tends to $+\\infty$ as $\\norm{x} \\to \\infty$.\nEquivalently, $p = \\operatorname{prox}_f(v)$ exactly when $v - p$ lies in the subdifferential $\\partial f(p)$.\nExample: let $C \\subseteq H$ be nonempty, closed and convex, and let $f$ be $0$ on $C$ and $+\\infty$ outside.\nThen $f(x) + \\tfrac12 \\norm{x - v}^2$ is finite only on $C$, so $\\operatorname{prox}_f(v)$ is the point of $C$ nearest to $v$, the projection of $v$ onto $C$.\nExample: for $f(x) = \\lambda \\abs{x}$ on $\\R$ with $\\lambda \u003e 0$, one finds $\\operatorname{prox}_f(v) = \\sgn(v) \\max(\\abs{v} - \\lambda, 0)$, the soft thresholding map.\nNon-example: let $f$ be $0$ on $\\set{-1, 1}$ and $+\\infty$ elsewhere on $\\R$, so $f$ is not convex.\nAt $v = 0$ both $-1$ and $1$ minimize $f(x) + \\tfrac12 x^2$, so no single proximal point is defined.\n",
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