LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-6-luna)

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The proximal bundle method returns within its finite trial budget

Statement

For a finite integer budget $N \geq 1$ , every invocation of Finite-memory-free proximal bundle method with serious and null steps returns one of its specified statuses after at most $N$ trials. Every trial point used by the procedure is uniquely defined unless an earlier oracle-failure or stationary-or-model-failure status has already returned.

Facts & Assumptions

Given: The nonempty closed convex feasible set, positive quadratic parameter, serious-step fraction, oracle rule, and finite trial budget of Finite-memory-free proximal bundle method with serious and null steps.

[F1]

A finite nonempty maximum-affine model plus the positive quadratic has a unique minimizer on the closed convex feasible set (The proximal bundle model subproblem has a unique minimizer).

[F2]

The initialization and trial oracle calls return immediately on failure, and each trial returns on nonpositive predicted decrease or otherwise completes one serious/null branch and increments its finite trial counter (Finite-memory-free proximal bundle method with serious and null steps).

Proof

1.1givenF1F2construct

If no cut is stored, the initialization oracle either fails and returns with zero completed trials or supplies one valid cut. Thereafter the bundle is finite and nonempty; at each trial [F1] supplies a unique candidate unless an earlier status has returned.

2.1F2step 1.1induction∎

Each trial either returns on oracle failure or $P_{k} \leq 0$ , or evaluates one candidate and completes exactly one serious/null branch, appends its cut, and advances the counter by one. Thus there are at most $N$ completed trials; if none returns earlier, the procedure returns budget-exhausted after trial $N$ .

Depends on

Used by

Nothing in the library uses this result yet.

Cited to discharge well-definedness by Finite-memory-free proximal bundle method with serious and null steps.

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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)