Gap-gradient methods for solving generalized mixed integer inverse optimization: an application to political gerrymandering
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866910486021275648 |
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| author | Smith, Ari J. Boutilier, Justin J. |
| author_facet | Smith, Ari J. Boutilier, Justin J. |
| contents | Inverse optimization has received much attention in recent years, but little literature exists for solving generalized mixed integer inverse optimization. We propose a new approach for solving generalized mixed-integer inverse optimization problems based on sub-gradient methods. We characterize when a generalized inverse optimization problem can be solved using sub-gradient methods and we prove that modifications to classic sub-gradient algorithms can return exact solutions in finite time. Our best implementation improves solution time by up to 90% compared to the best performing method from the literature. We then develop custom heuristic methods for graph-based inverse problems using a combination of graph coarsening and ensemble methods. Our heuristics are able to further reduce solution time by up to 52%, while still producing near-optimal solutions. Finally, we propose a new application domain - quantitatively identifying gerrymandering - for generalized inverse integer optimization. We apply our overall solution approach to analyze the congressional districts of the State of Iowa using real-world data. We find that the accepted districting marginally improves population imbalance at the cost of a significant increase in partisan efficiency gap. We argue that our approach can produce a more nuanced data-driven argument that a proposed districting should be considered gerrymandered. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_09457 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Gap-gradient methods for solving generalized mixed integer inverse optimization: an application to political gerrymandering Smith, Ari J. Boutilier, Justin J. Optimization and Control Inverse optimization has received much attention in recent years, but little literature exists for solving generalized mixed integer inverse optimization. We propose a new approach for solving generalized mixed-integer inverse optimization problems based on sub-gradient methods. We characterize when a generalized inverse optimization problem can be solved using sub-gradient methods and we prove that modifications to classic sub-gradient algorithms can return exact solutions in finite time. Our best implementation improves solution time by up to 90% compared to the best performing method from the literature. We then develop custom heuristic methods for graph-based inverse problems using a combination of graph coarsening and ensemble methods. Our heuristics are able to further reduce solution time by up to 52%, while still producing near-optimal solutions. Finally, we propose a new application domain - quantitatively identifying gerrymandering - for generalized inverse integer optimization. We apply our overall solution approach to analyze the congressional districts of the State of Iowa using real-world data. We find that the accepted districting marginally improves population imbalance at the cost of a significant increase in partisan efficiency gap. We argue that our approach can produce a more nuanced data-driven argument that a proposed districting should be considered gerrymandered. |
| title | Gap-gradient methods for solving generalized mixed integer inverse optimization: an application to political gerrymandering |
| topic | Optimization and Control |
| url | https://arxiv.org/abs/2406.09457 |