Footnote 4, p.10: $D(p,x)\ge(K/2)\|x-p\|^2$, so $D(p,x_t)\to0$ forces $x_t\to p$

Let $h$ be a distance-generating function with modulus $K$ on a compact convex set $\mathcal C$, with Bregman divergence $D$, and let $p\in\mathcal C$. Then:

1. $D(p,x)\ge\frac K2\|x-p\|^2$ for every $x$ in the prox-domain $\mathcal C_h$;
2. every sequence $x_t\in\mathcal C_h$ with $D(p,x_t)\to0$ converges to $p$.

Together with the reciprocity condition (RC), this shows that $x_t\to p$ if and only if $D(p,x_t)\to0$, which is how convergence of the energy is turned into convergence of the iterates.

Differences from the source:

• encoding: h : X → ℝ ∪ {∞} with dom h = C; ∂h(x) tested against all x′ ∈ X → h : E → ℝ with only its values on C used; the subgradient inequality tested for x′ ∈ C; reason: h = +∞ off C, so the inequality is automatic off C: the same subdifferential and prox-domain
• encoding: there exists a continuous selection ∇h of ∂h on dom ∂h → the selection is a parameter gradh with ContinuousOn and gradh x ∈ ∂h(x) on the prox-domain; reason: D and P are defined through the selection, so it must be fixed data
• encoding: K-strongly convex with modulus K (divided by in (A.5b), (40)) → 0 < Kc is part of IsDGF; reason: a strong-convexity modulus is positive; the paper divides by it