Pith Number

pith:PME675QG

pith:2026:PME675QGOZHYXQYNBTKRKYCSUG

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Accuracy and Relationships of Quadratic Models in Derivative-free Optimization

Lindon Roberts, Warren Hare, Yiwen Chen

Three quadratic models in derivative-free optimization all satisfy fully linear error bounds without assuming bounded model Hessians.

arxiv:2605.12819 v1 · 2026-05-12 · math.OC · cs.NA · math.NA

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Claims

C1strongest claim

We establish fully linear error bounds for all three models, removing the uniformly bounded model Hessian assumption required in existing MN analyses and deriving the first such results for the QS model. We further analyze Hessian approximation accuracy via directional error bounds, showing that all three models achieve fully quadratic accuracy along sample directions under a mild condition on the sample set.

C2weakest assumption

The mild geometric condition on the sample set required for directional fully quadratic accuracy, together with standard twice-differentiability of the objective function; if the sample geometry fails, the directional Hessian claims may not hold.

C3one line summary

The paper derives fully linear error bounds for minimum norm, minimum Frobenius norm, and quadratic generalized simplex derivative models, establishes directional fully quadratic Hessian accuracy under mild sample set conditions, and characterizes when the models coincide.

References

24 extracted · 24 resolved · 1 Pith anchors

[1] A matrix algebra approach to approximate Hessians , author=. IMA J. Numer. Anal. , volume=. 2024 , publisher= 2024

[2] An introduction to convexity, optimization, and algorithms , author=. 2023 , publisher= 2023

[3] Calculus identities for generalized simplex gradients: rules and applications , author=. SIAM J. Optim. , volume=. 2020 , publisher= 2020

[4] Q-fully quadratic modeling and its application in a random subspace derivative-free method , author=. Comput. Optim. Appl. , volume=. 2024 , publisher= 2024

[5] Chen, Y. and Hare, W. , title =. IMA J. Numer. Anal. , year =

Receipt and verification

First computed	2026-05-18T03:09:12.259097Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

7b09eff606764f8bc30d0cd5156052a188379c09cbe5ddf9b3739973ce5c00b0

Aliases

arxiv: 2605.12819 · arxiv_version: 2605.12819v1 · doi: 10.48550/arxiv.2605.12819 · pith_short_12: PME675QGOZHY · pith_short_16: PME675QGOZHYXQYN · pith_short_8: PME675QG

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Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/PME675QGOZHYXQYNBTKRKYCSUG \
  | jq -c '.canonical_record' \
  | python3 -c "import sys,json,hashlib; b=json.dumps(json.loads(sys.stdin.read()), sort_keys=True, separators=(',',':'), ensure_ascii=False).encode(); print(hashlib.sha256(b).hexdigest())"
# expect: 7b09eff606764f8bc30d0cd5156052a188379c09cbe5ddf9b3739973ce5c00b0

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "be7b9c5a641c71f0923733fa82b5378ae3a341b299b18706b0f0bbfadc122aae",
    "cross_cats_sorted": [
      "cs.NA",
      "math.NA"
    ],
    "license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
    "primary_cat": "math.OC",
    "submitted_at": "2026-05-12T23:23:27Z",
    "title_canon_sha256": "6b38b4372bbb1356215fa8892a435c4127c588b12fdd6413b71129c2b83e7d46"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2605.12819",
    "kind": "arxiv",
    "version": 1
  }
}