= Proximal gradient mapping
{title2=$G_\alpha(x)=\alpha^{-1}(x-\operatorname{prox}_{\alpha h}(x-\alpha\nabla f(x)))$}
= Proximal gradient map
{synonym}
For a <convex function> $h$ with a well-defined <proximal operator>, and a <differentiable function> $f$, the proximal gradient mapping is the displacement of one <proximal gradient method> update divided by its positive <step size> $\alpha$. Writing $p=x-\alpha G_\alpha(x)$, the <subgradient optimality condition> gives $G_\alpha(x)-\nabla f(x)\in\partial h(p)$. For <convex functions> $f,h$, its zeros are precisely the <global minimizers> of $f+h$, by the <subgradient optimality condition> and <subdifferential sum rule>. When $h=0$, it is the ordinary <gradient>.
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Target: convex-optimization.bigb, under Proximal gradient method.
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