arbiter
JournalsAccount

Bilevel Gradient Methods and the Morse Parametric Qualification Condition

Jérôme Bolte et al.

Mathematics of Operations Research2026https://doi.org/10.1287/moor.2025.0914preprint
AJG 3ABDC A
Weight
0.50

Abstract

We introduce the Morse parametric qualification condition for bilevel programming. Generic semialgebraic functions are Morse parametric in a piecewise sense. Thus, bilevel programs with a Morse parametric lower level constitute a relevant intermediate class between strongly convex and fully generic lower levels. In this framework, we study bilevel gradient algorithms with two strategies: the single-step multistep strategy, which involves a sequence of steps on the lower-level problems followed by one step on the upper-level problem, and a differentiable programming strategy that optimizes a smooth approximation of the bilevel problem. Although the first is shown to be a biased gradient method on the problem with rich properties, the second, which is inspired by metalearning applications, is less stable but offers simplicity and ease of implementation. Funding: This research was supported by the Agence Nationale de la Recherche [Grants ANR-19-PI3A-0004 and ANR-23-CE23-0012-01]. J. Bolte, Q.-T. Le, and E. Pauwels thank the AI Interdisciplinary Institute [ANITI funding], the Agence Nationale de la Recherche [France 2030 Program Grant ANR-23-IACL-0002], the Chair TRIAL, and the Air Force Office of Scientific Research [Grant FA8655-22-1-7012]. J. Bolte, E. Pauwels, and S. Vaiter acknowledge support from the ANR MAD. J. Bolte and E. Pauwels acknowledge support from the Agence Nationale de la Recherche [Grant ANR-17-EURE-0010]. Q.-T. Le is supported by the Agence Nationale de la Recherche [Grant ANR-23-IACL-0006]. E. Pauwels acknowledges support from the IUF. S. Vaiter thanks the Agence Nationale de la Recherche [Grant ANR-23-PEIA-0004] and the Chair 3IA BOGL [Grant ANR-23-IACL-0001].

Open via your library →

Cite this paper

https://doi.org/https://doi.org/10.1287/moor.2025.0914

Or copy a formatted citation

@article{jérôme2026,
  title        = {{Bilevel Gradient Methods and the Morse Parametric Qualification Condition}},
  author       = {Jérôme Bolte et al.},
  journal      = {Mathematics of Operations Research},
  year         = {2026},
  doi          = {https://doi.org/https://doi.org/10.1287/moor.2025.0914},
}

Paste directly into BibTeX, Zotero, or your reference manager.

Flag this paper

Bilevel Gradient Methods and the Morse Parametric Qualification Condition

Flags are reviewed by the Arbiter methodology team within 5 business days.

Evidence weight

0.50

Balanced mode · F 0.40 / M 0.15 / V 0.05 / R 0.40

F · citation impact0.50 × 0.4 = 0.20
M · momentum0.50 × 0.15 = 0.07
V · venue signal0.50 × 0.05 = 0.03
R · text relevance †0.50 × 0.4 = 0.20

† Text relevance is estimated at 0.50 on the detail page — for your query’s actual relevance score, open this paper from a search result.