Shifted-comparator Bregman bound ($D_J^w(u_{\alpha },u^{†})+\Vert K(u_{\alpha }-u^{†}+\alpha v){\Vert }^2/(2\alpha )\le D_J^w(u^{†}-\alpha v,u^{†})$)

Let $u^{†}$ be a least-squares solution and $w=K^{\ast }Kv\in \partial J(u^{†})$. For a minimizer $u_{\alpha }$ of $\Vert Ku-f{\Vert }^2/2+\alpha J(u)$, expand the objective about $u^{†}$. The normal equation for a linear inverse problem removes the data-residual cross term, while the Bregman distance definition givesCompare the minimizer with $u^{†}-\alpha v$, where the square vanishes, and divide by $\alpha$. The bound is informative when that comparator has finite penalty.