3
$\begingroup$

Let $f(x):\Re^n\rightarrow \Re$ be a proper and closed convex function. Its Moreau-Yosida regularization is defined as

$F(x)=\min_y\Big(f(y)+\frac{1}{2}\|y-x\|_2^2\Big)$

$\operatorname{Prox}_f(x)=\arg\min_y \Big(f(y)+\frac{1}{2}\|y-x\|_2^2\Big)$

Lots of literature say $F(x)$ is Lipschtiz continuous and give explicitly the expression of $\nabla F(x)$ involving $\operatorname{Prox}_f(x)$. But I have no idea how to calculate $\nabla F(x)$. Can anyone provide a straightforward method? I know Rockafellar's book gives a proof. But it assumes too much prior knowledge. I am wondering if there is a more elementary method to prove the Lipschtiz continuity and calculate its gradient.

  • 1
    $\mathbb R$ is a much more common notation for reals than $\mathfrak R$.2012-09-13
  • 0
    You mentioned the formula $\nabla f_{\mu}(x) = \frac{1}{\mu} ( x - \text{prox}_{\mu f}(x)$. In a lot of important applications, the prox operator of $f$ can be evaluated analytically. In other cases, you might have to solve a convex optimization problem in order to evaluate the prox operator of $f$. Vandenberghe's [236c notes](http://www.seas.ucla.edu/~vandenbe/236C/lectures/multmethods.pdf) (especially ch. 15 "multiplier methods") are a good resource for this topic.2013-10-09

2 Answers 2