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pith:5MGIFMV4

pith:2026:5MGIFMV4A5MN7SDWCBLB3SEED2

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Convergence Rates for $\ell_p$ Norm Minimization in Convex Vector Optimization

Mohammed Alshahrani

The Hausdorff approximation error converges at rate O(k^{2/(1-q)}) for every ℓ_p norm in convex vector optimization.

arxiv:2605.14324 v1 · 2026-05-14 · math.OC · cs.NA · math.NA

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4 Citations open

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The bundle contains the canonical record plus signed events. A mirror can host it anywhere and recompute the same current state with the deterministic merge algorithm.

Claims

C1strongest claim

We prove that the Hausdorff approximation error satisfies δ_H(P_k, A) = O(k^{2/(1-q)}) for every p ∈ (1,∞)

C2weakest assumption

The technique assumes the ambient space is R^q with its standard inner product structure, which enables the quadratic bound on hyperplane distance; this may not extend directly to non-Euclidean settings or infinite dimensions.

C3one line summary

The Hausdorff error for ℓ_p-norm based outer approximations in convex vector optimization converges at the optimal rate O(k^{2/(1-q)}) independently of p.

References

30 extracted · 30 resolved · 0 Pith anchors

[1] A Norm Minimization-Based Convex Vector Optimization Algorithm, 2022 · doi:10.1007/s10957-022-02045-8

[2] Convergence Analysis of a Norm Minimization-Based Convex Vector Optimization Algorithm, 2024 · doi:10.1137/23m1574580

[3] Asymptotic estimates for best and stepwise approximation of convex bodies II, 1993 · doi:10.1515/form.1993.5.521

[4] Asymptotic estimates for best and stepwise approximation of convex bodies I, 1993 · doi:10.1515/form.1993.5.281

[5] Asymptotic estimates for best and stepwise approximation of convex bodies III, 1997 · doi:10.1515/form.1997.9

Formal links

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Receipt and verification

First computed	2026-05-17T23:39:09.792220Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

eb0c82b2bc0758dfc87610561dc8841eaeac31f44ec5518ef4840f4bc427c951

Aliases

arxiv: 2605.14324 · arxiv_version: 2605.14324v1 · doi: 10.48550/arxiv.2605.14324 · pith_short_12: 5MGIFMV4A5MN · pith_short_16: 5MGIFMV4A5MN7SDW · pith_short_8: 5MGIFMV4

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Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/5MGIFMV4A5MN7SDWCBLB3SEED2 \
  | 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: eb0c82b2bc0758dfc87610561dc8841eaeac31f44ec5518ef4840f4bc427c951

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "771d7597400db5dbcc60de14a6be36af5146d66e0eed8af795e9cff0c65267e3",
    "cross_cats_sorted": [
      "cs.NA",
      "math.NA"
    ],
    "license": "http://creativecommons.org/licenses/by/4.0/",
    "primary_cat": "math.OC",
    "submitted_at": "2026-05-14T03:39:06Z",
    "title_canon_sha256": "7412982852ae2812bf0483ab37f77c5c5bd508d7d7eeb9d557db13dee2fa0121"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2605.14324",
    "kind": "arxiv",
    "version": 1
  }
}