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Let $H$ be a Hilbert space (e.g Euclidean $\mathbb R^n$), and fix a proper convex function $f:H \to (-\infty,+\infty]$. Given any $t \ge 0$, let $P_t:H \to H$ be the proximal operator of $f$ at level $t$, given by $$ P_{t}(x) := \arg\min_{y \in H}\|x-y\|_H^2 + t f(y). $$

Fix any $x \in H$, and define $d_x:\mathbb R_+ \to \mathbb R_+$ by $d_x(t):= \|P_{t}(x)-x\|_H^2$.

Question. Is it true that $d_x$ a non-decreasing function ?

Example. If, $f(x) := \|x\|_H^2/2$, then $P_{t}(x) \equiv x / (1+t)$, and so $d_x(t) = t^2\|x\|_H^2/(1+t)^2$, which is certainly non-decreasing in $t$.

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    $\begingroup$ plz. check lemma 4 of arxiv.org/pdf/1109.0258.pdf -- a similar proof should help here. $\endgroup$
    – Suvrit
    Commented Feb 13, 2023 at 2:40
  • $\begingroup$ Thanks! Indeed, in the Euclidean case $H=\mathbb R^n$, your Lemma 4 answer's my question in the affirmative (by taking $z=0$). The argument though written for the Euclidean case carries over to the Hilbert setting. $\endgroup$
    – dohmatob
    Commented Feb 13, 2023 at 9:02

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