Academic Journal
Recovering Dantzig–Wolfe Bounds by Cutting Planes.
| Τίτλος: | Recovering Dantzig–Wolfe Bounds by Cutting Planes. |
|---|---|
| Συγγραφείς: | Chen, Rui1 (AUTHOR) rui.chen@cornell.edu, Günlük, Oktay2 (AUTHOR) oktay.gunluk@cornell.edu, Lodi, Andrea3 (AUTHOR) andrea.lodi@cornell.edu |
| Πηγή: | Operations Research. Mar/Apr2025, Vol. 73 Issue 2, p1128-1142. 15p. |
| Θεματικοί όροι: | *Integer programming, *Prices, Mixed integer linear programming, Assignment problems (Programming), Knapsack problems |
| Περίληψη: | Leveraging Dantzig–Wolfe Decomposition in the Original Variable Space for Mixed-Integer Programming Dantzig–Wolfe decomposition has been extensively applied to solve large-scale mixed-integer programs with decomposable structures, leading to exact solution approaches, such as branch and price. However, these approaches would require solving the problem in an extended variable space and are not readily present in off-the-shelf solvers. In "Recovering Dantzig–Wolfe Bounds by Cutting Planes," Chen, Günlük, and Lodi propose a computational effective approach for generating cutting planes from Dantzig–Wolfe decomposition to enhance branch and cut in the space of original variables. The proposed approach requires a relatively small number of cutting planes to recover the strength of the Dantzig–Wolfe dual bound and should be easy to implement in general-purpose mixed-integer programming solvers. The authors show that these cutting planes typically lead to a formulation with lower dual degeneracy and hence, a better computational performance than naïve approaches, such as the objective function cut. Dantzig–Wolfe (DW) decomposition is a well-known technique in mixed-integer programming (MIP) for decomposing and convexifying constraints to obtain potentially strong dual bounds. We investigate cutting planes that can be derived using the DW decomposition algorithm and show that these cuts can provide the same dual bounds as DW decomposition. More precisely, we generate one cut for each DW block, and when combined with the constraints in the original formulation, these cuts imply the objective function cut one can simply write using the DW bound. This approach typically leads to a formulation with lower dual degeneracy that consequently has a better computational performance when solved by standard MIP solvers in the original space. We also discuss how to strengthen these cuts to improve the computational performance further. We test our approach on the multiple knapsack assignment problem and the temporal knapsack problem, and we show that the proposed cuts are helpful in accelerating the solution time without the need to implement branch and price. Funding: This work was supported by the Office of Naval Research [Grant N00014-21-1-2575]. Supplemental Material: The e-companion is available at https://doi.org/10.1287/opre.2023.0048. [ABSTRACT FROM AUTHOR] |
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| Items | – Name: Title Label: Title Group: Ti Data: Recovering Dantzig–Wolfe Bounds by Cutting Planes. – Name: Author Label: Authors Group: Au Data: <searchLink fieldCode="AR" term="%22Chen%2C+Rui%22">Chen, Rui</searchLink><relatesTo>1</relatesTo> (AUTHOR)<i> rui.chen@cornell.edu</i><br /><searchLink fieldCode="AR" term="%22Günlük%2C+Oktay%22">Günlük, Oktay</searchLink><relatesTo>2</relatesTo> (AUTHOR)<i> oktay.gunluk@cornell.edu</i><br /><searchLink fieldCode="AR" term="%22Lodi%2C+Andrea%22">Lodi, Andrea</searchLink><relatesTo>3</relatesTo> (AUTHOR)<i> andrea.lodi@cornell.edu</i> – Name: TitleSource Label: Source Group: Src Data: <searchLink fieldCode="JN" term="%22Operations+Research%22">Operations Research</searchLink>. Mar/Apr2025, Vol. 73 Issue 2, p1128-1142. 15p. – Name: Subject Label: Subject Terms Group: Su Data: *<searchLink fieldCode="DE" term="%22Integer+programming%22">Integer programming</searchLink><br />*<searchLink fieldCode="DE" term="%22Prices%22">Prices</searchLink><br /><searchLink fieldCode="DE" term="%22Mixed+integer+linear+programming%22">Mixed integer linear programming</searchLink><br /><searchLink fieldCode="DE" term="%22Assignment+problems+%28Programming%29%22">Assignment problems (Programming)</searchLink><br /><searchLink fieldCode="DE" term="%22Knapsack+problems%22">Knapsack problems</searchLink> – Name: Abstract Label: Abstract Group: Ab Data: Leveraging Dantzig–Wolfe Decomposition in the Original Variable Space for Mixed-Integer Programming Dantzig–Wolfe decomposition has been extensively applied to solve large-scale mixed-integer programs with decomposable structures, leading to exact solution approaches, such as branch and price. However, these approaches would require solving the problem in an extended variable space and are not readily present in off-the-shelf solvers. In "Recovering Dantzig–Wolfe Bounds by Cutting Planes," Chen, Günlük, and Lodi propose a computational effective approach for generating cutting planes from Dantzig–Wolfe decomposition to enhance branch and cut in the space of original variables. The proposed approach requires a relatively small number of cutting planes to recover the strength of the Dantzig–Wolfe dual bound and should be easy to implement in general-purpose mixed-integer programming solvers. The authors show that these cutting planes typically lead to a formulation with lower dual degeneracy and hence, a better computational performance than naïve approaches, such as the objective function cut. Dantzig–Wolfe (DW) decomposition is a well-known technique in mixed-integer programming (MIP) for decomposing and convexifying constraints to obtain potentially strong dual bounds. We investigate cutting planes that can be derived using the DW decomposition algorithm and show that these cuts can provide the same dual bounds as DW decomposition. More precisely, we generate one cut for each DW block, and when combined with the constraints in the original formulation, these cuts imply the objective function cut one can simply write using the DW bound. This approach typically leads to a formulation with lower dual degeneracy that consequently has a better computational performance when solved by standard MIP solvers in the original space. We also discuss how to strengthen these cuts to improve the computational performance further. We test our approach on the multiple knapsack assignment problem and the temporal knapsack problem, and we show that the proposed cuts are helpful in accelerating the solution time without the need to implement branch and price. Funding: This work was supported by the Office of Naval Research [Grant N00014-21-1-2575]. Supplemental Material: The e-companion is available at https://doi.org/10.1287/opre.2023.0048. [ABSTRACT FROM AUTHOR] – Name: AbstractSuppliedCopyright Label: Group: Ab Data: <i>Copyright of Operations Research is the property of INFORMS: Institute for Operations Research & the Management Sciences and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.) |
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| RecordInfo | BibRecord: BibEntity: Identifiers: – Type: doi Value: 10.1287/opre.2023.0048 Languages: – Code: eng Text: English PhysicalDescription: Pagination: PageCount: 15 StartPage: 1128 Subjects: – SubjectFull: Integer programming Type: general – SubjectFull: Prices Type: general – SubjectFull: Mixed integer linear programming Type: general – SubjectFull: Assignment problems (Programming) Type: general – SubjectFull: Knapsack problems Type: general Titles: – TitleFull: Recovering Dantzig–Wolfe Bounds by Cutting Planes. Type: main BibRelationships: HasContributorRelationships: – PersonEntity: Name: NameFull: Chen, Rui – PersonEntity: Name: NameFull: Günlük, Oktay – PersonEntity: Name: NameFull: Lodi, Andrea IsPartOfRelationships: – BibEntity: Dates: – D: 01 M: 03 Text: Mar/Apr2025 Type: published Y: 2025 Identifiers: – Type: issn-print Value: 0030364X Numbering: – Type: volume Value: 73 – Type: issue Value: 2 Titles: – TitleFull: Operations Research Type: main |
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