{"paper":{"title":"On the Nature of Regularity Assumptions in Bilevel Optimization with Constrained Lower-level Problem","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"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.","cross_cats":[],"primary_cat":"math.OC","authors_text":"Chang He, Mingyi Hong, Shuzhong Zhang, Xiaotian Jiang","submitted_at":"2026-05-14T05:52:04Z","abstract_excerpt":"In this paper, we study the regularity assumptions commonly adopted in bilevel optimization with constrained lower-level problems, including the linear independence constraint qualification, the strict complementary slackness condition, and the second-order sufficient condition. These conditions are typically required to hold for the lower-level problem at every upper-level variable $x$. We first show that 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 perturbati"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"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.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"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.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"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.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"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.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"a8059078787759f6c82b079900a9fba6702e4ae168bda8d71230854671bff82d"},"source":{"id":"2605.14409","kind":"arxiv","version":1},"verdict":{"id":"0c936ba0-86a1-4188-839a-4b6b7f441f8a","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-15T01:53:42.973453Z","strongest_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.","one_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.","pipeline_version":"pith-pipeline@v0.9.0","weakest_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.","pith_extraction_headline":"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."},"references":{"count":91,"sample":[{"doi":"","year":2012,"title":"Stackelberg network pricing games , author=. Algorithmica , volume=. 2012 , publisher=","work_id":"1a6349fb-8306-4e3a-a0d5-202a9c9c2be9","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2002,"title":"Ba. A. Journal of optimization theory and applications , volume=. 2002 , publisher=","work_id":"107a316f-c451-4506-9d14-a74f62db9510","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":null,"title":"Advances in Neural Information Processing Systems , volume=","work_id":"2d54371b-4dda-4473-8db5-0daae3750905","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":null,"title":"Advances in Neural Information Processing Systems , volume=","work_id":"a72f5916-e7fb-42d7-b747-17e5b9de24da","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2015,"title":"Journal of the Operations Research Society of Japan , volume=","work_id":"e8cabd04-bcde-4005-be9f-5b3d5ec978c1","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":91,"snapshot_sha256":"fb270ad7ac510df078811c31ebb2bd761bd178bbfd16691e77c46660f3753c11","internal_anchors":0},"formal_canon":{"evidence_count":2,"snapshot_sha256":"ef2e5069e4eb46a187d33be02d881e28af88e30ef06e183cf6dbcf904ba21753"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}