Local boundedness of the proximal gradient mapping ($\Vert G_{\alpha }(x)\Vert \le \Vert \nabla f(x)+w\Vert ,w\in \partial h(x)$)

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 $(v-w{)}^T(p-x)\ge 0$ for any fixed $w\in \partial h(x)$. HenceThe 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 $L{\alpha }^2\Vert \nabla f(x)+w{\Vert }^2/2$. Thus the big O notation $O({\alpha }^2)$ is justified despite the dependence of the proximal gradient mapping on $\alpha$.

Target: convex-optimization.bigb, under Coordinate descent.