Dual Space

Given a closed, convex and proper function $f:\mathbf{R}^n\to\mathbf{R}$, its proximity operator $\operatorname{prox}_f:\mathbf{R}^n\to\mathbf{R}^n$ is defined as \[ \operatorname{prox}_f(y)=\operatorname{argmin}_{x\in\mathbf{R}^n}\{f(x)+\tfrac{1}{2}\|x-y\|_2^2\}\mbox{.} \] It is well-known that the scaled Euclidean norm $\lambda\|\cdot\|_2$ with $\lambda>0$ has a closed-form proximity operator, \[ \operatorname{prox}_{\lambda\|\cdot\|_2}(y)=\left(1-\frac{\lambda}{\max\{\|y\|_2,\lambda\}}\right)y\mbox{.} \] This can be derived using the Moreau identity, or by using optimality conditions. I’ll spare the details since… — read more