Forward subgradient step ($F_{auf}$)

For a convex function and $\tau >0$, the forward step is the set-valued explicit update $F_{\tau f}(x)=\{x-\tau p:p\in \partial f(x)\}$. It reduces to the usual gradient step when differentiable. In contrast, a backward step uses a subgradient at the new point and is the proximal operator for a proper closed convex function.