= Shifted-comparator Bregman bound
{title2=$D_J^w(u_\alpha,u^\dagger)+\|K(u_\alpha-u^\dagger+\alpha v)\|^2/(2\alpha)\leq D_J^w(u^\dagger-\alpha v,u^\dagger)$}
Let $u^\dagger$ be a <least-squares solution> and $w=K^*Kv\in\partial J(u^\dagger)$. For a minimizer $u_\alpha$ of $\|Ku-f\|^2/2+\alpha J(u)$, expand the objective about $u^\dagger$. The <normal equation for a linear inverse problem> removes the data-residual cross term, while the <Bregman distance> definition gives
$$
\mathcal F_\alpha(u)-\mathcal F_\alpha(u^\dagger)=\frac12\|K(u-u^\dagger+\alpha v)\|^2+\alpha D_J^w(u,u^\dagger)-\frac{\alpha^2}2\|Kv\|^2.
$$
Compare the minimizer with $u^\dagger-\alpha v$, where the square vanishes, and divide by $\alpha$. The bound is informative when that comparator has finite penalty.
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