Bilevel Gradient Methods and the Morse Parametric Qualification Condition

• 成果类型: Article; Early Access
• 署名作者: Bolte, Jerome; Le, Quoc-Tung; Pauwels, Edouard; Vaiter, Samuel
• 署名单位: Communaute d'universites et etablissements de Toulouse (Comue); Universite Toulouse 1 Capitole; Toulouse School of Economics; Centre National de la Recherche Scientifique (CNRS); Communaute Universite Grenoble Alpes; Universite Grenoble Alpes (UGA); Inria; Institut National Polytechnique de Grenoble; Centre National de la Recherche Scientifique (CNRS); Universite Cote d'Azur
• 刊物名称: MATHEMATICS OF OPERATIONS RESEARCH
• ISSN/ISSBN: 0364-765X; 1526-5471
• DOI: 10.1287/moor.2025.0914
• 发表日期: 2026-04-08
• 关键词: bilevel optimization; Morse qualification condition; automatic differentiation; minimization; PROGRAMS

摘要

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.