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Vanishing aggregate-model error controls the proximal candidate

Statement

Let $F : R^{n} \to (- \infty, + \infty]$ be proper, closed, and convex, with nonempty closed convex domain $C = dom F$ . Fix $t > 0$ , a center $x \in C$ , and a finite affine model $M \leq F$ on $C$ . Let $z$ minimize $M (y) + ∥ y - x ∥_{2}^{2} / (2 t)$ over $C$ , set $e = F (z) - M (z) \geq 0$ , and let $p = {prox}_{t F} (x)$ . Then $∥ z - p ∥_{2}^{2} \leq 4 t e .$ In particular, a sequence of such model errors $e_{k} \to 0$ forces the model candidates to approach their exact proximal points in norm.

Facts & Assumptions

Given: Bundle subproblem existence The proximal bundle model subproblem has a unique minimizer, proximal mapping Proximal mapping, resolvent/existence Existence, resolvent characterization, and firm nonexpansiveness of the proximal map, and quadratic growth A quadratic regularization has quadratic growth from its minimizer.

[F1]

$p$ is the unique minimizer of $Q_{F} (y) = F (y) + ∥ y - x ∥_{2}^{2} / (2 t)$ (Proximal mapping, Existence, resolvent characterization, and firm nonexpansiveness of the proximal map).

[F2]

The convex function $Q_{F}$ obeys $Q_{F} (z) \geq Q_{F} (p) + ∥ z - p ∥_{2}^{2} / (4 t)$ (A quadratic regularization has quadratic growth from its minimizer).

Proof

1.1givenF1algebra

Put $Q_{M} (y) = M (y) + ∥ y - x ∥_{2}^{2} / (2 t)$ . Since $z$ minimizes $Q_{M}$ over $C$ and $p \in C$ , $Q_{M} (z) \leq Q_{M} (p)$ . The pointwise underestimator gives $Q_{M} (p) \leq Q_{F} (p)$ , while $Q_{F} (z) = Q_{M} (z) + e$ . Hence $Q_{F} (z) \leq Q_{F} (p) + e$ .

2.1F2step 1.1algebra∎

Apply [F2] to $z$ and combine with step 1.1: $Q_{F} (p) + ∥ z - p ∥_{2}^{2} / (4 t) \leq Q_{F} (z) \leq Q_{F} (p) + e$ . Subtracting the common finite value and multiplying by $4 t > 0$ proves the inequality.

Depends on

Used by

Proximal bundle accumulation under vanishing aggregate-model error Theorem

Dependency tree · two levels

39 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Mateo Díaz and Benjamin Grimmer, Optimal Convergence Rates for the Proximal Bundle Method, §§2–3 (standard reference, not scraped)
Dominikus Noll, Olivier Prot, and Aude Rondepierre, A proximity control algorithm to minimize nonsmooth and nonconvex functions, §§2–4 (standard reference, not scraped)