= Local boundedness of the proximal gradient mapping
{title2=$\|G_\alpha(x)\|\leq\|\nabla f(x)+w\|,\quad w\in\partial h(x)$}
Fix $x$ with nonempty <subdifferential> $\partial h(x)$ for a <convex function> $h$. If $p=x-\alpha G_\alpha(x)$ and $v=G_\alpha(x)-\nabla f(x)\in\partial h(p)$, <monotonicity of a convex subdifferential> gives $(v-w)^T(p-x)\geq0$ for any fixed $w\in\partial h(x)$. Hence
$$
\|G_\alpha(x)\|^2\leq(\nabla f(x)+w)^TG_\alpha(x)\leq\|\nabla f(x)+w\|\,\|G_\alpha(x)\|.
$$
The <Cauchy-Schwarz inequality> proves the bound uniformly for $\alpha>0$. If $f$ has an $L$-<Lipschitz gradient>, this makes the remainder in the <proximal gradient mapping> descent bound at most $L\alpha^2\|\nabla f(x)+w\|^2/2$. Thus the <big O notation> $O(\alpha^2)$ is justified despite the dependence of the <proximal gradient mapping> on $\alpha$.
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Target: convex-optimization.bigb, under Coordinate descent.
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