ExampleConstruction: Literature-sourcedVerification: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-6-luna)

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

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Nelder–Mead can stop on a nonminimizing flat plateau

Example

Define the continuous objective on $R$ by $f (x) = 0$ for $x \leq 2$ , $f (x) = 2 - x$ for $2 < x < 3$ , and $f (x) = - 1$ for $x \geq 3$ . Run Finite-budget Nelder–Mead simplex procedure from the one-dimensional simplex $(0, 1)$ with diameter tolerance $0.1$ , spread tolerance $0.01$ , and call budget $14$ . It returns tolerance-met after four shrink updates, with simplex $(0, 1 / 16)$ , returned best point $0$ , and objective spread $0$ , although the global minimum is $- 1$ on $[3, \infty)$ .

Facts & Assumptions

Given: Exact branch comparisons, tie rules, shrink rule, tolerance test, and call accounting Finite-budget Nelder–Mead simplex procedure.

[A1]

The stated piecewise formula defines $f$ on all of $R$ ; its pieces agree at $2$ and $3$ .

[F1]

The algorithm keeps the lower-slot vertex on objective ties, uses strict inside-contraction acceptance, and shrinks toward the best vertex (Finite-budget Nelder–Mead simplex procedure).

Verification

1.1givenA1F1algebra

The two initial values at $0, 1$ are both $0$ ; the slot tie rule makes $0$ best and $1$ worst. At a simplex $(0, h)$ with $0 < h \leq 1$ , reflection is $- h$ , with value $0 = f (h)$ . Since the reflected value ties the worst value, the inside-contraction branch applies; its point is $h / 2$ and also has value $0$ , so strict inside-contraction acceptance fails and shrink produces $(0, h / 2)$ .

2.1givenstep 1.1algebra∎

Starting at $h = 1$ , four repetitions give $h = 1 / 16 = 0.0625 < 0.1$ and the spread remains $0 < 0.01$ . Initial evaluations use two calls and each iteration uses reflection, inside contraction, and a fresh shrink evaluation, so the total is $2 + 4 \cdot 3 = 14$ calls. All evaluated points lie in $[- 1, 1] \subset (- \infty, 2]$ , where the objective is flat; meanwhile $f (x) \geq - 1$ everywhere and equals $- 1$ for $x \geq 3$ . Thus the prescribed finite tolerance is met at a non-global minimizer.

Depends on

Finite-budget Nelder–Mead simplex procedure

Used by

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Sources

Tamara G. Kolda, Robert Michael Lewis, and Virginia Torczon, Optimization by Direct Search, Nelder–Mead discussion (standard reference, not scraped)
K. I. M. McKinnon, Convergence of the Nelder–Mead Simplex Method to a Nonstationary Point, §§2–3 (standard reference, not scraped)