Pipeline-generated

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.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

✓ 3 results · all verified · 3 also independently AI-judged

Every result on this page is machine-checked by a proof checker and read in full and owner-audited; the judge is an additional, independent cross-model AI review of the proofs; all 3 also cleared it.

Subgradient bundle and derivative free optimization — Examples

1 · Prerequisites

2 · Summary

Examples

These examples calculate a null step followed by a serious step for the max-affine kink $f (x) = ∣ x ∣$ , run direct search on a black-box quadratic using only function values, and exhibit Nelder–Mead meeting both finite tolerances on a flat plateau that is not globally minimizing. Their exact updates and evaluation counts distinguish what a finite run reports from stationarity or global-optimality guarantees.

3 · Logical flowchart

4 · Definitions, theorems and proofs

None yet.

5 · Examples, counterexamples and false statements

ExampleConstruction: Literature-sourcedVerification: AI-adaptedprecheck passjudge pass (gpt-6-luna)Open item page →

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.

ExampleConstruction: Literature-sourcedVerification: AI-adaptedprecheck passjudge pass (gpt-6-luna)Open item page →

A black-box quadratic simulator fit using direct search

Example

Treat the simulator response $f (θ) = (θ - 2)^{2}$ as a black-box evaluation, with no derivative supplied. Run Finite-budget deterministic direct search with positive-spanning polling from $θ_{0} = 0$ using ordered polls $D = (+ 1, - 1)$ , $Δ_{0} = 1$ , cap $Δ_{\max} = 2$ , forcing $ρ (Δ) = Δ^{2} / 10$ , increase factor $2$ , decrease factor $1 / 2$ , and poll budget $N = 3$ . The successive centers are $0, 1, 1, 2$ ; the third poll reaches the exact simulator fit $θ = 2$ , and the finite run returns status budget-exhausted.

Facts & Assumptions

Given: Positive-spanning direct-search polling and forcing rule Finite-budget deterministic direct search with positive-spanning polling.

[F1]

In one dimension $D = (+ 1, - 1)$ is positive spanning; the acceptance test compares only function values and the stated forcing threshold (Finite-budget deterministic direct search with positive-spanning polling).

Verification

1.1givenF1algebra

At center $0$ , $f (0) = 4$ and the first poll point $1$ has $f (1) = 1 < 4 - 1 / 10$ , so the first poll succeeds at $1$ and the mesh grows to $2$ .

2.1givenF1step 1.1algebra∎

At center $1$ with mesh $2$ , the ordered trials $3, - 1$ have values $1, 9$ , neither below $f (1) - 4 / 10 = 3 / 5$ ; the second poll is unsuccessful and halves the mesh to $1$ . At center $1$ with mesh $1$ , the first trial $2$ has value $0 < 1 - 1 / 10$ , so the third poll succeeds at $2$ . The budget is now exhausted; the observed exact minimizer is returned without a derivative-based certificate.

ExampleConstruction: Literature-sourcedVerification: AI-adaptedprecheck passjudge pass (gpt-6-luna)Open item page →

A max-affine kink produces one null and one serious bundle step

Example

Run two trials of Finite-memory-free proximal bundle method with serious and null steps on $f (x) = ∣ x ∣ = \max {x, - x}$ over $C = R$ , with $t = 2$ , $m_{s} = 1 / 2$ , and initial center $x_{0} = 1$ . At $1$ choose subgradient $1$ ; at the first trial point $- 1$ choose subgradient $- 1$ ; at $0$ choose subgradient $0$ . The first trial is null and the second is serious, with exact trial points $- 1$ and $0$ respectively.

Facts & Assumptions

Given: The serious/null algorithm and subgradient definition Finite-memory-free proximal bundle method with serious and null steps, Subgradients and the subdifferential of a convex function.

[A1]

For $λ \in [0, 1]$ , $f ((1 - λ) x + λ y) = \max {(1 - λ) x + λ y, - (1 - λ) x - λ y} \leq (1 - λ) \max {x, - x} + λ \max {y, - y}$ , because each affine branch is bounded by the right side; hence $f$ is convex.

[F1]

For every $y$ , $∣ y ∣ \geq y$ , $∣ y ∣ \geq - y$ , and $∣ y ∣ \geq 0$ , so the chosen slopes $1, - 1, 0$ support $f$ at $1, - 1, 0$ respectively (Subgradients and the subdifferential of a convex function).

[F2]

The finite maximum-affine model plus its positive quadratic has a unique minimizer (The proximal bundle model subproblem has a unique minimizer).

Verification

1.1givenA1F1F2algebra

The initial cut is $m_{0} (y) = y$ . The model objective is $y + (y - 1)^{2} / 4 = (y + 1)^{2} / 4$ , so its unique minimizer is $z_{0} = - 1$ . Then $M_{0} (z_{0}) = - 1$ , $P_{0} = f (1) - M_{0} (- 1) = 2$ , and actual decrease is $f (1) - f (- 1) = 0 < m_{s} P_{0} = 1$ . Thus this is a null step and the cut at $- 1$ is $m_{1} (y) = - y$ .

2.1givenF1F2step 1.1algebra∎

The new model is $M_{1} (y) = \max {y, - y} = ∣ y ∣$ . For $y \geq 0$ , $∣ y ∣ + (y - 1)^{2} / 4 = (y + 1)^{2} / 4 \geq 1 / 4$ , with equality only at $y = 0$ . For $y \leq 0$ , its difference from $1 / 4$ is $y (y - 6) / 4 \geq 0$ because both factors are nonpositive, with equality only at $y = 0$ . Thus its unique minimizer is $z_{1} = 0$ , and $P_{1} = f (1) - M_{1} (0) = 1$ , while actual decrease is $1 \geq m_{s} P_{1} = 1 / 2$ . The second step is serious and sets the center to $0$ .

Subgradient bundle and derivative free optimization — Examples

1 · Prerequisites

2 · Summary

Examples

3 · Logical flowchart

4 · Definitions, theorems and proofs

5 · Examples, counterexamples and false statements

Nelder–Mead can stop on a nonminimizing flat plateau

Example

Facts & Assumptions

Verification

A black-box quadratic simulator fit using direct search

Example

Facts & Assumptions

Verification

A max-affine kink produces one null and one serious bundle step

Example

Facts & Assumptions

Verification

Sources