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Coordinated cutting plane generation via multi-objective separation

Coordinated cutting plane generation via multi-objective separation
Coordinated cutting plane generation via multi-objective separation
In cutting plane methods, the question of how to generate the “best possible” set of cuts is both central and crucial. We propose a lexicographic multi-objective cutting plane generation scheme that generates, among all the maximally violated valid inequalities of a given family, an inequality that is undominated and maximally diverse w.r.t. the cuts that were previously found. By optimizing a diversity measure, we introduce a form of coordination between successive cuts. Our focus is on valid inequalities with 0–1 coefficients in the left-hand side and a constant right-hand side, which encompasses several families of valid inequalities. As cut diversity measure, we consider an aggregate of the 1-norm distances w.r.t. the normal vectors of the previous cuts. In this case, our lexicographic multi-objective separation problem reduces to the standard separation problem with different values for the objective function coefficients. The impact of our coordinated cutting plane generation scheme is assessed in a pure cutting plane setting when separating stable set and cut set inequalities for, respectively, the max clique and min Steiner tree problems. Compared to the standard separation of undominated maximally violated cuts, we close the same fraction of the duality gap in a considerably smaller number of rounds and cuts. The potential of our scheme is also indicated by the results obtained in a cut-and-branch setting for max clique, where cut coordination allows for a substantial reduction, on average, of the number of branch-and-bound nodes.
0025-5610
87-110
Amaldi, Edoardo
eefad18b-86c1-4a8f-a931-23e8ddd59d6e
Coniglio, Stefano
03838248-2ce4-4dbc-a6f4-e010d6fdac67
Gualandi, Stefano
8eaa785d-388a-45e5-bba2-6bda706ab2ae
Amaldi, Edoardo
eefad18b-86c1-4a8f-a931-23e8ddd59d6e
Coniglio, Stefano
03838248-2ce4-4dbc-a6f4-e010d6fdac67
Gualandi, Stefano
8eaa785d-388a-45e5-bba2-6bda706ab2ae

Amaldi, Edoardo, Coniglio, Stefano and Gualandi, Stefano (2014) Coordinated cutting plane generation via multi-objective separation. Mathematical Programming, 143 (1-2), 87-110. (doi:10.1007/s10107-012-0596-x).

Record type: Article

Abstract

In cutting plane methods, the question of how to generate the “best possible” set of cuts is both central and crucial. We propose a lexicographic multi-objective cutting plane generation scheme that generates, among all the maximally violated valid inequalities of a given family, an inequality that is undominated and maximally diverse w.r.t. the cuts that were previously found. By optimizing a diversity measure, we introduce a form of coordination between successive cuts. Our focus is on valid inequalities with 0–1 coefficients in the left-hand side and a constant right-hand side, which encompasses several families of valid inequalities. As cut diversity measure, we consider an aggregate of the 1-norm distances w.r.t. the normal vectors of the previous cuts. In this case, our lexicographic multi-objective separation problem reduces to the standard separation problem with different values for the objective function coefficients. The impact of our coordinated cutting plane generation scheme is assessed in a pure cutting plane setting when separating stable set and cut set inequalities for, respectively, the max clique and min Steiner tree problems. Compared to the standard separation of undominated maximally violated cuts, we close the same fraction of the duality gap in a considerably smaller number of rounds and cuts. The potential of our scheme is also indicated by the results obtained in a cut-and-branch setting for max clique, where cut coordination allows for a substantial reduction, on average, of the number of branch-and-bound nodes.

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More information

Accepted/In Press date: 11 September 2012
e-pub ahead of print date: 18 October 2012
Published date: February 2014
Organisations: Operational Research

Identifiers

Local EPrints ID: 410176
URI: http://eprints.soton.ac.uk/id/eprint/410176
ISSN: 0025-5610
PURE UUID: 199ac953-2bac-4c91-bab9-942572e062ba
ORCID for Stefano Coniglio: ORCID iD orcid.org/0000-0001-9568-4385

Catalogue record

Date deposited: 06 Jun 2017 04:02
Last modified: 16 Mar 2024 04:24

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Contributors

Author: Edoardo Amaldi
Author: Stefano Gualandi

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