On disjunction convex hulls by lifting
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866912097781153792 |
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| author | Qu, Yushan Lee, Jon |
| author_facet | Qu, Yushan Lee, Jon |
| contents | We study the natural extended-variable formulation for the disjunction of $n+1$ polytopes in $\mathbb{R}^d$. We demonstrate that the convex hull $D$ in the natural extended-variable space $\mathbb{R}^{d+n}$ is given by full optimal big-M lifting (i) when $d\leq 2$ (and that it is not generally true for $d\geq 3$), and also (ii) under some technical conditions, when the polytopes have a common facet-describing constraint matrix, for arbitrary $d\geq 1$ and $n\geq 1$. We give a broad family of examples with $d\geq 3$ and $n=1$, where the convex hull is not described after employing all full optimal big-M lifting inequalities, but it is described after one round of MIR inequalities. Additionally, we give some general results on the polyhedral structure of $D$, and we demonstrate that all facets of $D$ can be enumerated in polynomial time when $d$ is fixed. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2407_15244 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On disjunction convex hulls by lifting Qu, Yushan Lee, Jon Optimization and Control Combinatorics We study the natural extended-variable formulation for the disjunction of $n+1$ polytopes in $\mathbb{R}^d$. We demonstrate that the convex hull $D$ in the natural extended-variable space $\mathbb{R}^{d+n}$ is given by full optimal big-M lifting (i) when $d\leq 2$ (and that it is not generally true for $d\geq 3$), and also (ii) under some technical conditions, when the polytopes have a common facet-describing constraint matrix, for arbitrary $d\geq 1$ and $n\geq 1$. We give a broad family of examples with $d\geq 3$ and $n=1$, where the convex hull is not described after employing all full optimal big-M lifting inequalities, but it is described after one round of MIR inequalities. Additionally, we give some general results on the polyhedral structure of $D$, and we demonstrate that all facets of $D$ can be enumerated in polynomial time when $d$ is fixed. |
| title | On disjunction convex hulls by lifting |
| topic | Optimization and Control Combinatorics |
| url | https://arxiv.org/abs/2407.15244 |