https://en.wikipedia.org/wiki/Moreau_envelope

$$ M_f(v)\equiv \inf_{x\in \mathcal X}\left(f(x)+\frac 12\|x-v\|_2^2\right) $$

$$ \nabla M_{\lambda f}(v)=\frac 1\lambda (x-\operatorname{prox}_{\lambda f}(x)) $$

⇒ $x_{k+1}=\operatorname{prox}{\lambda f}(x_k)=x-\lambda \nabla M{\lambda f}(x)$

• Proximal operator of a (proper, lower semi-continuous convex) function f

  $$ \operatorname{prox}{f}(v)=\argmin{x\in \mathcal X}\left( f(x)+\frac 12\|x-v\|_{\mathcal X}^2 \right) $$

  对于指示函数，近端算子等价于投影算子

• Fenchel’s Duality theorem