On the multisource hyperplanes location problem to fitting set of points
•Multifacility location of hyperplanes.•Mixed integer programming formulations.•Set partitioning formulation.•Branch and Price algorithm. In this paper we study the problem of locating a given number of hyperplanes minimizing an objective function of the closest distances from a set of points. We pr...
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| Published in: | Computers & operations research Vol. 128; p. 105124 |
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| Format: | Journal Article |
| Language: | English |
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01.04.2021
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| ISSN: | 0305-0548, 1873-765X |
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| Abstract | •Multifacility location of hyperplanes.•Mixed integer programming formulations.•Set partitioning formulation.•Branch and Price algorithm.
In this paper we study the problem of locating a given number of hyperplanes minimizing an objective function of the closest distances from a set of points. We propose a general framework for the problem in which norm-based distances between points and hyperplanes are aggregated by means of ordered median functions. A compact Mixed Integer Linear (or Non Linear) programming formulation is presented for the problem and also an extended set partitioning formulation with a huge number of variables is derived. We develop a column generation procedure embedded within a branch-and-price algorithm for solving the problem by adequately performing its preprocessing, pricing and branching. We also analyze geometrically the optimal solutions of the problem, deriving properties which are exploited to generate initial solutions for the proposed algorithms. Finally, the results of an extensive computational experience are reported. The issue of scalability is also addressed showing theoretical upper bounds on the errors assumed by replacing the original datasets by aggregated versions. |
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| AbstractList | •Multifacility location of hyperplanes.•Mixed integer programming formulations.•Set partitioning formulation.•Branch and Price algorithm.
In this paper we study the problem of locating a given number of hyperplanes minimizing an objective function of the closest distances from a set of points. We propose a general framework for the problem in which norm-based distances between points and hyperplanes are aggregated by means of ordered median functions. A compact Mixed Integer Linear (or Non Linear) programming formulation is presented for the problem and also an extended set partitioning formulation with a huge number of variables is derived. We develop a column generation procedure embedded within a branch-and-price algorithm for solving the problem by adequately performing its preprocessing, pricing and branching. We also analyze geometrically the optimal solutions of the problem, deriving properties which are exploited to generate initial solutions for the proposed algorithms. Finally, the results of an extensive computational experience are reported. The issue of scalability is also addressed showing theoretical upper bounds on the errors assumed by replacing the original datasets by aggregated versions. |
| ArticleNumber | 105124 |
| Author | Ponce, D. Blanco, V. Japón, A. Puerto, J. |
| Author_xml | – sequence: 1 givenname: V. surname: Blanco fullname: Blanco, V. email: vblanco@ugr.es organization: IEMath-GR, Universidad de Granada, Spain – sequence: 2 givenname: A. surname: Japón fullname: Japón, A. email: ajapon1@us.es organization: IMUS, Universidad de Sevilla, Spain – sequence: 3 givenname: D. orcidid: 0000-0003-3380-6601 surname: Ponce fullname: Ponce, D. email: dponce@us.es organization: IMUS, Universidad de Sevilla, Spain – sequence: 4 givenname: J. surname: Puerto fullname: Puerto, J. email: puerto@us.es organization: IMUS, Universidad de Sevilla, Spain |
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| Keywords | Column generation 90B85 90C30 90C11 Hyperplanes location 52C35 Mixed Integer Non Linear programming |
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| Snippet | •Multifacility location of hyperplanes.•Mixed integer programming formulations.•Set partitioning formulation.•Branch and Price algorithm.
In this paper we... |
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| SubjectTerms | Column generation Hyperplanes location Mixed Integer Non Linear programming |
| Title | On the multisource hyperplanes location problem to fitting set of points |
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