Optimistic Bilevel Optimization with Composite Lower-Level Problem
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866910012747546624 |
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| author | Solla, Mattia Royset, Johannes O. |
| author_facet | Solla, Mattia Royset, Johannes O. |
| contents | This paper introduces a novel double regularization scheme for bilevel optimization problems whose lower-level problem is composite and convex, but not necessarily strongly convex, in the lower-level variable. The analysis focuses on the primal-dual solution mapping of the regularized lower-level problem and exploits its properties to derive an almost-everywhere formula for the gradient of the regularized hyper-objective under mild assumptions. The paper then establishes conditions under which the hyper-objective of the actual problem is well defined and shows that its gradient can be approximated by the gradient of the regularized hyper-objective. Building on these results, a gradient sampling-based algorithm computes approximately stationary points of the regularized hyper-objective, and we prove its convergence to stationary points of the actual problem. Two numerical examples from machine learning demonstrate the proposed approach. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_05417 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Optimistic Bilevel Optimization with Composite Lower-Level Problem Solla, Mattia Royset, Johannes O. Optimization and Control 90C31 (Primary) 90C30, 90C26, 90C56 (Secondary) This paper introduces a novel double regularization scheme for bilevel optimization problems whose lower-level problem is composite and convex, but not necessarily strongly convex, in the lower-level variable. The analysis focuses on the primal-dual solution mapping of the regularized lower-level problem and exploits its properties to derive an almost-everywhere formula for the gradient of the regularized hyper-objective under mild assumptions. The paper then establishes conditions under which the hyper-objective of the actual problem is well defined and shows that its gradient can be approximated by the gradient of the regularized hyper-objective. Building on these results, a gradient sampling-based algorithm computes approximately stationary points of the regularized hyper-objective, and we prove its convergence to stationary points of the actual problem. Two numerical examples from machine learning demonstrate the proposed approach. |
| title | Optimistic Bilevel Optimization with Composite Lower-Level Problem |
| topic | Optimization and Control 90C31 (Primary) 90C30, 90C26, 90C56 (Secondary) |
| url | https://arxiv.org/abs/2602.05417 |