Pith Number

pith:2RTRNKAF

pith:2026:2RTRNKAF6QUUFFXRWOJY62WOHQ

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On the Nature of Regularity Assumptions in Bilevel Optimization with Constrained Lower-level Problem

Chang He, Mingyi Hong, Shuzhong Zhang, Xiaotian Jiang

Requiring lower-level regularity conditions at every upper-level point in bilevel optimization is non-prevalent, as structural invariants cannot be made consistent by small perturbations.

arxiv:2605.14409 v1 · 2026-05-14 · math.OC

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Claims

C1strongest claim

the requirement that these conditions hold at every upper-level variable x is strong, in the sense that it is non-prevalent: there exist problems for which no sufficiently small perturbation of the lower-level objective and constraints can make the conditions hold at every x.

C2weakest assumption

The rigidity theorems assume that the lower-level problem is defined by smooth functions and that the structural invariants (e.g., active-set signatures) are well-defined and constant when the regularity conditions hold; this may fail for non-smooth or degenerate lower-level problems not covered by the counterexamples.

C3one line summary

Requiring LICQ/SCS/SOSC everywhere in bilevel optimization is non-prevalent and rigid, while holding almost everywhere is prevalent, but the distinction introduces fundamental difficulties.

References

91 extracted · 91 resolved · 0 Pith anchors

[1] Stackelberg network pricing games , author=. Algorithmica , volume=. 2012 , publisher= 2012

[2] Ba. A. Journal of optimization theory and applications , volume=. 2002 , publisher= 2002

[3] Advances in Neural Information Processing Systems , volume=

[4] Advances in Neural Information Processing Systems , volume=

[5] Journal of the Operations Research Society of Japan , volume= 2015

Formal links

2 machine-checked theorem links

Receipt and verification

First computed	2026-05-17T23:39:07.387165Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

d46716a805f4294296f1b3938f6ace3c1ba73750084587e3f456ae329102c025

Aliases

arxiv: 2605.14409 · arxiv_version: 2605.14409v1 · doi: 10.48550/arxiv.2605.14409 · pith_short_12: 2RTRNKAF6QUU · pith_short_16: 2RTRNKAF6QUUFFXR · pith_short_8: 2RTRNKAF

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Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/2RTRNKAF6QUUFFXRWOJY62WOHQ \
  | 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: d46716a805f4294296f1b3938f6ace3c1ba73750084587e3f456ae329102c025

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "7f3b02ca9550728ad01c4248622e233ddb6745b926fa8ad36507d1adb0c0a67d",
    "cross_cats_sorted": [],
    "license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
    "primary_cat": "math.OC",
    "submitted_at": "2026-05-14T05:52:04Z",
    "title_canon_sha256": "cf0b67444fa1a650d168747a0c69442117b2d60fe1b7c9ba4d52df23de85d748"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2605.14409",
    "kind": "arxiv",
    "version": 1
  }
}