Bilevel linear optimization belongs to NP and admits polynomial-size KKT-based reformulations
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arXiv
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866911324516122624 |
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| author | Buchheim, Christoph |
| author_facet | Buchheim, Christoph |
| contents | It is a well-known result that bilevel linear optimization is NP-hard. In many publications, reformulations as mixed-integer linear optimization problems are proposed, which suggests that the decision version of the problem belongs to NP. However, to the best of our knowledge, a rigorous proof of membership in NP has never been published, so we close this gap by reporting a simple but not entirely trivial proof. A related question is whether a large enough "big M" for the classical KKT-based reformulation can be computed efficiently, which we answer in the affirmative. In particular, our big M has polynomial encoding length in the original problem data. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2307_06639 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Bilevel linear optimization belongs to NP and admits polynomial-size KKT-based reformulations Buchheim, Christoph Optimization and Control It is a well-known result that bilevel linear optimization is NP-hard. In many publications, reformulations as mixed-integer linear optimization problems are proposed, which suggests that the decision version of the problem belongs to NP. However, to the best of our knowledge, a rigorous proof of membership in NP has never been published, so we close this gap by reporting a simple but not entirely trivial proof. A related question is whether a large enough "big M" for the classical KKT-based reformulation can be computed efficiently, which we answer in the affirmative. In particular, our big M has polynomial encoding length in the original problem data. |
| title | Bilevel linear optimization belongs to NP and admits polynomial-size KKT-based reformulations |
| topic | Optimization and Control |
| url | https://arxiv.org/abs/2307.06639 |