Pith Number

pith:255WVFSU

pith:2026:255WVFSUMKFFSS4SGJ3K4IEPGI

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Nonsmooth Set-Gradient Ascent to the Pareto Front via Layered Hypervolume and Magnitude Indicators

Michael T.M. Emmerich

Layered weighting of nondomination layers gives set-gradient ascent directions that improve the first Pareto front without compensation from deeper layers.

arxiv:2605.13468 v1 · 2026-05-13 · math.OC · cs.NA · cs.NE · math.NA

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Claims

C1strongest claim

The method gives ascent directions to nondominated and dominated points while preventing deeper layers from compensating for deterioration of the first front, with an exact gradient formula for the magnitude indicator derived as a linear combination of hypervolume gradients of projected shadow sets.

C2weakest assumption

That rapidly decreasing weights on successive nondomination layers suffice to isolate improvement of the first front and that chamberwise Lipschitz continuity on bounded sets holds for the finite-epsilon surrogates used in practice.

C3one line summary

Nonsmooth gradient ascent on layered hypervolume and magnitude indicators moves sets to the Pareto front.

References

21 extracted · 21 resolved · 1 Pith anchors

[1] F. H. Clarke,Optimization and Nonsmooth Analysis, Classics in Applied Mathematics 5, SIAM, Philadelphia, 1990 1990

[2] Hypervolume-Based Multiobjective Optimization: Theoretical Foundations and Practical Implications, 2012 · doi:10.1016/j.tcs.2011.03.012

[3] Direct Multisearch for Multiobjective Opti- mization, 2011 · doi:10.1137/10079731x

[4] Deb,Multi-Objective Optimization Using Evolutionary Algorithms, Wiley, Chichester, 2001 2001

[5] Multi-objective Optimization by Uncrowded Hypervolume Gradient Ascent, 2020 · doi:10.1007/978-3-030-58115-2_13

Receipt and verification

First computed	2026-05-18T02:44:41.589087Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

d77b6a9654628a594b923276ae208f321fd88e1fc2a9f554a69affaad6d654c5

Aliases

arxiv: 2605.13468 · arxiv_version: 2605.13468v1 · doi: 10.48550/arxiv.2605.13468 · pith_short_12: 255WVFSUMKFF · pith_short_16: 255WVFSUMKFFSS4S · pith_short_8: 255WVFSU

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Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/255WVFSUMKFFSS4SGJ3K4IEPGI \
  | 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: d77b6a9654628a594b923276ae208f321fd88e1fc2a9f554a69affaad6d654c5

Canonical record JSON

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    "abstract_canon_sha256": "d9043667ea4dcb8d09300620ea86d03693ff07b5511e568e8c290cf01437ed50",
    "cross_cats_sorted": [
      "cs.NA",
      "cs.NE",
      "math.NA"
    ],
    "license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
    "primary_cat": "math.OC",
    "submitted_at": "2026-05-13T12:55:22Z",
    "title_canon_sha256": "d86bdcab9cf1ba92cfa7ed846c70b53e43408df3ea71e4890c0a0f0fb51256aa"
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  "schema_version": "1.0",
  "source": {
    "id": "2605.13468",
    "kind": "arxiv",
    "version": 1
  }
}