# A concise proof of Moreau's characterization of proximal operators

Kunal Narayan Chaudhury

► To cite this version:

Kunal Narayan Chaudhury. A concise proof of Moreau's characterization of proximal operators. Indian Institute of Science. 2025, pp.1-2. hal-05038838v3

HAL Id: hal-05038838

https://hal.science/hal-05038838v3

Submitted on 21 May 2025

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_1.png" /></figure>
# A CONCISE PROOF OF MOREAU'S CHARACTERIZATION OF PROXIMAL OPERATORS

KUNAL N. CHAUDHURY

**ABSTRACT.** We present a concise proof of Moreau's characterization of proximal operators [4]. While the core ideas of the proof can already be found in the five-decade-old paper, our approach is more streamlined, consolidating parts of the original argument using now-standard concepts from convex optimization and monotone operator theory. This might be particularly useful to researchers, as the original paper is in French and, to our knowledge, no translation is currently available. We also discuss a corollary of Moreau's theorem for linear operators, which appears in some recent works on image reconstruction.

## 1. MOREAU'S THEOREM

Let $X$ be a real finite-dimensional inner product space and $\Gamma_0(X)$ denote the set of functions $f: X \to \mathbb{R} \cup (-\infty, +\infty]$ that are proper, convex, and lower semicontinuous [1]. Namely, $f \in \Gamma_0(X)$ iff $f(x_0) < +\infty$ for some $x_0 \in X$,

$$ \forall \theta \in (0, 1), \forall x_1, x_2 \in X, \quad f(\theta x_1 + (1-\theta)x_2) \le \theta f(x_1) + (1-\theta)f(x_2), $$

and

$$ \forall \bar{x} \in X, \forall \{x_k\} \to \bar{x} : \quad f(\bar{x}) \le \liminf_{k \to \infty} f(x_k). $$

The proximal operator of $f \in \Gamma_0(X)$ is defined as

$$ \operatorname{prox}_f : X \to X, \quad \operatorname{prox}_f(x) = \operatorname*{argmin}_{z \in X} (q(x-z) + f(z)) \quad (x \in X). $$

where we use $q$ to denote the quadratic function $q(x) = (1/2)\|x\|^2$.

**Theorem 1 (Moreau, [4]).** Let $T$ be an operator on $X$. Then $T = \operatorname{prox}_f$ for some $f \in \Gamma_0(X)$ if and only if $T$ is nonexpansive and $T = \nabla\phi$ for some differentiable $\phi \in \Gamma_0(X)$.

If $T$ is linear, then there is a particularly simple characterization, which was used in some recent works on image reconstruction [3, 5, 6].

**Corollary 2.** Let $W$ be a linear operator on $X$. Then $W = \operatorname{prox}_f$ for some $f \in \Gamma_0(X)$ if and only if $W$ is symmetric and its eigenvalues are in $[0, 1]$.

## 2. PROOF

As with the original proof of Theorem 1, we will use Fenchel duality. Recall that the Fenchel dual of $f \in \Gamma_0(X)$ is defined as

$$ f^* : X \to \mathbb{R} \cup [-\infty, +\infty], \quad f^*(\xi) = \sup_{x \in X} \langle \xi, x \rangle - f(x) \quad (\xi \in X). $$

<footnote>contact: kunal@iisc.ac.in.</footnote>
For any $f: X \to \mathbb{R} \cup (-\infty, \infty]$, the dual $f^*$ takes values in $(-\infty, +\infty]$, provided $f$ is proper. Thus, $f^*$ is convex and lower semicontinuous for any proper $f$, but $f^*$ itself need not be proper.

We will also need the concept of the Moreau envelope of a function $f \in \Gamma_0(X)$ given by

$$M_f : X \to \mathbb{R}, \quad M_f(x) = \min_{z \in X} q(x-z) + f(z) \quad (x \in X).$$

We need the following basic results about Fenchel duality and the relation between the proximal operator and the Moreau envelope; e.g., see [1, 2].

**Lemma 3.** Let $f \in \Gamma_0(X)$. Then

(1) $f^* \in \Gamma_0(X)$ and $(f^*)^* = f$.

(2) $(f+q)^* = M_{f^*}$.

(3) $M_f \in \Gamma_0(X) \cap C^1(X)$ and $\nabla M_f = J_{f^*}$.

(4) $\text{prox}_f$ is nonexpansive.

*Proof of Theorem 1.* Suppose $T = \text{prox}_f$. Then $\text{prox}_f$ is nonexpansive by Lemma 3. Moreover, if we set $\phi = M_{f^*}$, then by Lemma 3, $\phi \in \Gamma_0(X)$ is differentiable, and

$$\nabla\phi = \nabla M_{f^*} = \text{prox}_f = T.$$

In the other direction, suppose $T$ is nonexpansive and $T = \nabla\phi$ for some $\phi \in \Gamma_0(X)$. Define

$$g(x) = q(x) - \phi(x) \quad (x \in X).$$

We claim that $g \in \Gamma_0(X)$. Obviously, $g$ is proper and lower semicontinuous. Since $g$ is differentiable, to show that $g$ is convex, it suffices to check that $\nabla g$ is monotone, i.e., for all $x, y \in X$,

(1) $$\langle x - y, \nabla g(x) - \nabla g(y) \rangle \ge 0.$$

Indeed, since $\nabla g(x) = x - T(x)$, we have from the Cauchy-Schwarz inequality,

$$\begin{aligned} \langle x - y, \nabla g(x) - \nabla g(y) \rangle &= \|x - y\|^2 - \langle x - y, T(x) - T(y) \rangle \\ &\ge \|x - y\|^2 - \|x - y\| \|T(x) - T(y)\|. \end{aligned}$$

As $T$ is nonexpansive, the quantity on the right-hand side is nonnegative, giving us (1) and establishing our claim.

We next write $\phi(x) = q(x) - g(x)$, and compute

$$\phi^*(\xi) = \sup_{x \in X} \langle \xi, x \rangle - \phi(x) = \sup_{x \in X} \langle \xi, x \rangle - q(x) + g(x).$$

Since $g \in \Gamma_0(X)$, $g = (g^*)^*$ by Lemma 3. Thus, we can write

$$\begin{aligned} \phi^*(\xi) &= \sup_{x \in X} \langle \xi, x \rangle - q(x) + \sup_{\zeta \in X} \langle \zeta, x \rangle - g^*(\zeta) \\ &= \sup_{\zeta \in X} -g^*(\zeta) + \sup_{x \in X} \langle \xi + \zeta, x \rangle - q(x) \\ &= \sup_{\zeta \in X} -g^*(\zeta) + q(\xi + \zeta) \\ &= q(\xi) + \sup_{\zeta \in X} \langle \xi, \zeta \rangle - h(\zeta) \\ &= q(\xi) + h^*(\xi), \end{aligned}$$
where $h := g^* - q$. We cannot say that $h$ is convex, but $h^*$ is lower semicontinuous and convex. Moreover, $h^* = \phi^* - q$, so that $h^*$ is proper. Thus, $h^* \in \Gamma_0(X)$ and $k := (h^*)^* \in \Gamma_0(X)$.

As $\phi \in \Gamma_0(X)$, we have from Lemma 3 that $\phi = (\phi^*)^* = (h^* + q)^* = M_k$. Consequently,

$$T = \nabla\phi = \nabla M_k = J_{k^*}.$$

Thus, we have shown that $T = \text{prox}_f$, where $f := k^* \in \Gamma_0(X)$, completing the proof. □

*Proof of Corollary 2.* Suppose that $W$ is symmetric and its eigenvalues are in $[0, 1]$. Define

$$\phi(x) = \frac{1}{2} \langle x, Wx \rangle \quad (x \in X).$$

Then $\phi \in \Gamma_0(X)$ and $Wx = \nabla\phi(x)$. Consequently, it follows from Theorem 1 that $W = \text{prox}_f$ for some $f \in \Gamma_0(X)$.

In the other direction, assume that $W = \text{prox}_f$ for some $f \in \Gamma_0(X)$. We know from Theorem 1 that $Wx = \nabla\phi(x)$ for some convex differentiable function $\phi$. In particular, this means that $\phi$ is twice continuously differentiable and $\nabla^2\phi(x) = W$. We can thus conclude from Clairaut's theorem [7] that $\nabla^2\phi(x)$, and hence $W$, is symmetric. In particular, $W$ has real eigenvalues. Furthermore, the eigenvalues are nonnegative as $\nabla^2\phi(x)$ is positive semidefinite. On the other hand, as $W = \text{prox}_f$ is nonexpansive, the eigenvalues cannot be greater than 1. □

### 3. ACKNOWLEDGEMENT

A big thanks to Ruturaj Gavaskar for his patient and careful preparation of the English translation of [4].

### REFERENCES

1. H. H. Bauschke and P. L. Combettes, *Convex analysis and monotone operator theory in Hilbert spaces*, Springer, 2017.

2. Amir Beck, *First-order methods in optimization*, SIAM, 2017.

3. R. G. Gavaskar, C. D. Athalye, and K. N. Chaudhury, *On plug-and-play regularization using linear denoisers*, IEEE Transactions on Image Processing **30** (2021), 4802–4813.

4. J.-J. Moreau, *Proximité et dualité dans un espace hilbertien*, Bulletin de la Société mathématique de France **93** (1965), 273–299.

5. A. Sinha and K. N. Chaudhury, *On the strong convexity of pnp regularization using linear denoisers*, IEEE Signal Processing Letters **31** (2024), 2790–2794.

6. S. Sreehari et al., *Plug-and-play priors for bright field electron tomography and sparse interpolation*, IEEE Transactions on Computational Imaging **2** (2016), no. 4, 408–423.

7. T. Tao, *Analysis II*, Springer, 2016.