# Questions tagged [proximal-operators]

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).

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### Proximal point operator of maximum of functions

I want to know whether the proximal point operator of a function with a special structure can be expressed in terms of simpler proximal point operators. Setup: For a proper, convex and lower ...
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### Results on convergence and runtime rates of proximal algorithms

Are there any known results on the convergence rates and computational runtime of proximal algorithms? I'm interested in finding out how well they scale with increasing number of input dimensions, but ...
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### Same result in every iterations from subgradient and proximal gradient method.

I'm trying to implement the subgradient method and proximal gradient method with constant stepsize for the lasso problem but the result for the subgradient method and proximal gradient is almost ...
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### Manipulating a linear form into a proximal operator form

Given an expression: $$\arg\min_x -\alpha p^TAx + f(x)$$ I'd like to define $v$ such that: $$\text{prox}_{\sigma f}(v) = \arg\min_x 0.5 \|x - v\|_2^2 + \sigma f(x) = \arg \min_x -\alpha p^TAx + f(x)$$ ...
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### Proof for basic properties of proximal operators

I am reading the paper "Proximal algorithms" by N. Parikh and S. Boyd, and I found interesting the basic properties of proximal operators. However, I can't prove the equivalence for the ...
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### To prove an inequality related to the proximal operator (or non-expensive operator)

I am seeking assistance in proving an inequality that I believe holds for a specific mathematical concept involving the proximal operator based on a proper convex function. The inequality is as ...
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### is a function which all stationary points are global minima prox-regular for some $\alpha$>0?

Let $f$ be a function which all stationary points are global minima. This type of functions are also known as invex. It means, there exits $\eta(x,y)$ such that $f(x)-f(y) \geq \zeta_{y}^{T}\eta(x,y)$ ...
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### Proximal operator of squared $\ell_1$-norm

For any $a \in \mathbb R^d$ and $t \ge 0$, let $p_t(a)$ be the unique minimizer of $f_t(x;a) := \|x-a\|_2^2 + t\|x\|_1^2$ over $x \in \mathbb R^d$. Question. Is there an analytic formula for $p_t(a)$ ?...
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### Proximal operator of the $L^1$ norm, constrained to $x\ge0$

A standard variational problem (arising, e.g., in imaging) reads $$\operatorname{argmin}_x \frac{1}{2}\Vert Ax - y\Vert ^2 + \Vert x\Vert_1$$ where the $1$-norm serves as a sparsifying regularizer. ...
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Let f$,g$ be proper closed convex functions, $u=(y,z)$, and the problem: $$$$\begin{split} \min_{x,u} \ &f(x) + g(u) \\ \text{s.t. } & Ax + Bu = c \end{split}$$$$ ...
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### Why does $f:\mathbb{R}^n\rightarrow\mathbb{R}\cup\{\infty\}$ closed, proper, convex imply that $f(x)+\|Ax-b\|^2$ attains a minimum?

I'm currently reading the paper "Distributed Optimization and Statistical Learning via the Alternating Direction Method of Multipliers" by Boyd et al. about the alternating direction method ...
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### Proof for the orthogonal projection onto the L-infinity $L_{\infty}$ norm ball for the complex-valued

Problem: Prove orthogonal projection onto the $L_{\infty}$ norm ball, for $x \in \mathbb{C}^N$ and $\eta \geq 0$, is \begin{align} P_{B_{\| \cdot \|_{\infty} \leq \eta}} &= \left[\operatorname{...
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### Connection between proximal mapping and strong convexity

I need some helps to solve this proof, any guidance will be greatly appreciated. Suppose $\mathbf{x}_k$=$Prox_{\eta}\,g(\mathbf{y}_k-\eta\nabla f(\mathbf{y}_k))$, prove that: \begin{align*} g(\mathbf{...
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### Proximal operator intuition

I'm studying proximal operators. But there is something that disturbs me quite a lot and can't find an answer to it. If ℎ is a closed convex function, then the proximal operator of ℎ (with parameter 𝑡...
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### Proximal Point of the absolute value function

Absolute value function is defined as follows: \begin{equation*} f(x) = |x| = \left\{ \begin{array}{ll} x & \mbox{if } x \geq 0 \\ -x & \mbox{if } x < 0 \end{array} \right. \end{...
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### Different definitions of the proximal map? What is going on here

I am really confused about the proximal operator. My teacher has lectured and posted notes where she defines the proximal operator like this: I am trying to get some intuition for this weird function....
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### Proximal operator of the function $w \mapsto \max_{i=1}^k a_i^Tw$, for fixed $a_1,\ldots,a_k \in \mathbb R^n$
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 $F$ ? ...