Use this tag in question related to the Proximal Operator / Proximal Mapping. It might also be used in question about Proximal Gradient Method and Alternating Direction Method of Multipliers (ADMM).

Dedicated to the Proximal Operator.

Use this tag in question related to the Proximal Operator / Proximal Mapping. It might also be used in question about Proximal Gradient Method and Alternating Direction Method of Multipliers (ADMM).

Dedicated to the Proximal Operator.

155 questions

votes

Let $a_1,\ldots,a_k \in \mathbb R^n$ and consider the convex function $F:\mathbb R^n \to \mathbb R$ defined by $F(w) := \max_{i=1}^k a_i^\top w $.
Question. What is the proximal operator of $F$ ? That is, for $\lambda > 0$, find an a closed-form…

dohmatob

- 8,341
- 16
- 52

votes

For any convex, proper and closed function $f$ and for any $x$, the Moreau decomposition states that
$$Prox_f(x)+Prox_{f^*}(x)=x,$$
where $f^*$ is the conjugate function of $f$ and $Prox_f$ is the proximal operator of $f$ defined as…

Regev Cohen

- 141
- 5

votes

The proximal mapping is $$\text{prox}_{\eta, g}(w) = \arg\min_z \Big[\frac{1}{2\eta} ||z-w||_2^2 + g(z) \Big]$$
Now we want to consider the function $$g(w) = \sum_{i=1}^d \frac{\lambda}{\alpha} \ln(1 + \alpha|w_j|)$$
What is the proximal mapping…

sedrick

- 1,184
- 5
- 16

votes

For a convex function $f: \mathbb{R} \to \mathbb{R}$, define the Moreau envelope
$$f_{\mu}(x) = \inf_y \left\{ f(y) + \frac{1}{2\mu} (x-y)^2 \right\}$$
and the proximal operator
$$\text{prox}_{\mu f}(x) = \text{arg min}_y \left\{ f(y) +…

user3294195

- 469
- 2
- 13

votes

I would like the proximal operator of the following function:
\begin{align}
f(x) &= \lambda_1 \|x\|_1 + \lambda_2 \|x\|_{2,1} \\
&= \lambda_1 \|x\|_1 + \lambda_2 \sum_g \|x_g\|_2
\end{align}
where $x$ is the concatenation of all $x_g$,…

NicNic8

- 6,492
- 3
- 17
- 31

votes

Consider a proximable function $f$ where the proximal operator is defined as follows,
$$\operatorname{prox}_{\lambda f}(x) = \arg \min_z \frac{1}{2\lambda}\left\| z - x \right\|_2^2 + f(z)$$
$\lambda \geq 0$. With an additional constraint the…

Kumar

- 282
- 1
- 10

votes

I have problem in following form:
Let $X\in R^{N\times M}$ denote feature matrix for $M$ features and $Y\in R^{N\times T}$ be response matrix for $N$ data points over $T$ variables.
I have function $$f(X) + h(W)$$
where $f(W)=||Y-XW||_F^2$
I solve…

user2806363

- 245
- 2
- 16

votes

I need to find the proximal operator for $f(x) = \lambda \sqrt{x^2 + a^2}$, where $x \in \mathbb{R}$ and $a, \lambda >0$ are some constants. Is there a closed form for it? I tried to derive the operator in a straightforward way, i.e., using its…

A. Coifman

- 31
- 3

votes

I'm trying to solve an optimization problem of the form
$$\text{minimize } \; f(x) + \|x\|_\infty$$
where $x$ ranges over all of $\mathbb{R}^n$ and $f:\mathbb{R}^n \to \mathbb{R}$ is a nice, smooth, convex, differentiable function. Notice the…

D.W.

- 2,695
- 1
- 22
- 47

votes

I am reading the ADMM paper by S. Boyd et al - Distributed Optimization and Statistical Learning via the Alternating Direction Method of Multipliers.
I'm interested in implementing a L1 regularized feature wise distributed multinomial logistic…

theaNO

- 31
- 1

votes

I am looking for ways to compute the subderivative of $ ||Au||_{L^{\infty}} $, as I want to solve the minimization problem of
\begin{equation} \min\limits_u \quad \lambda ||Au||_{L^{\infty}} + \frac{1}{2} || u - f ||^2_{L^{2}} \quad.
\end{equation}…

mirrormere

- 55
- 1
- 5

votes

I hope to solve this problem.
$$\min \quad \left\| CX \right\|_{1} $$
$$ \text{s.t.}\quad AX=b, X >0 $$
where $C \in \mathbb{R}^{m \times m}$, $X \in \mathbb{R}^{m \times n}$, $A \in \mathbb{R}^{k \times m}$, $b \in \mathbb{R}^{k \times n}$. $C$ is…

Xia

- 111
- 12

votes

I have a problem in hand for which I need to compute the proximal operator of the composite function $ {\left\| \mbox{Hankel} (x) \right\|}_{\ast} $ where $ x \in \mathbb R^N $ and $ \left\| \cdot \right\|_{\ast} $ denotes the matrix nuclear…

mmn61

- 31
- 1

votes

Let ${C \subset \mathbb{R}^n}$ be closed, convex, and nonempty. How might one show that the proximal mapping of the indicator function of $C$ is in fact the projection operator on to $C$?

gdoug

- 143
- 5

votes

How to solve the following optimization problem in $ X \in \mathbb{C}^{N \times M} $?
\begin{equation}
\hat{X} = \arg \min_{X} \frac{1}{2} {\left\| X - Y \right\|}_{F}^{2} + \lambda {\left\| X \right\|}_{\ast}
\end{equation}
Where $ {\left\|…

Huayu Zhang

- 503
- 4
- 15