Multi-objective unconstrained combinatorial optimization: a polynomial bound on the number of extreme supported solutions
The multi-objective unconstrained combinatorial optimization problem (MUCO) can be considered as an archetype of a discrete linear multi-objective optimization problem. It can be interpreted as a specific relaxation of any multi-objective combinatorial optimization problem with linear sum objective...
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| Veröffentlicht in: | Journal of global optimization 2019-07, Vol.74 (3), p.495-522 |
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| creator | Schulze, Britta Klamroth, Kathrin Stiglmayr, Michael |
| description | The multi-objective unconstrained combinatorial optimization problem (MUCO) can be considered as an archetype of a discrete linear multi-objective optimization problem. It can be interpreted as a specific relaxation of any multi-objective combinatorial optimization problem with linear sum objective function. While its single criteria analogon is analytically solvable, MUCO shares the computational complexity issues of most multi-objective combinatorial optimization problems: intractability and NP-hardness of the
ε
-constraint scalarizations. In this article interrelations between the supported points of a MUCO problem, arrangements of hyperplanes and a weight space decomposition, and zonotopes are presented. Based on these interrelations and a result by Zaslavsky on the number of faces in an arrangement of hyperplanes, a polynomial bound on the number of extreme supported solutions can be derived, leading to an exact polynomial time algorithm to find all extreme supported solutions. It is shown how this algorithm can be incorporated into a solution approach for multi-objective knapsack problems. |
| doi_str_mv | 10.1007/s10898-019-00745-6 |
| format | Article |
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ε
-constraint scalarizations. In this article interrelations between the supported points of a MUCO problem, arrangements of hyperplanes and a weight space decomposition, and zonotopes are presented. Based on these interrelations and a result by Zaslavsky on the number of faces in an arrangement of hyperplanes, a polynomial bound on the number of extreme supported solutions can be derived, leading to an exact polynomial time algorithm to find all extreme supported solutions. It is shown how this algorithm can be incorporated into a solution approach for multi-objective knapsack problems.</description><identifier>ISSN: 0925-5001</identifier><identifier>EISSN: 1573-2916</identifier><identifier>DOI: 10.1007/s10898-019-00745-6</identifier><language>eng</language><publisher>New York: Springer US</publisher><subject>Algorithms ; Combinatorial analysis ; Computer Science ; Extreme values ; Hyperplanes ; Mathematical programming ; Mathematics ; Mathematics and Statistics ; Mechanical properties ; Multiple objective analysis ; Operations Research/Decision Theory ; Optimization ; Polynomials ; Real Functions ; Traveling salesman problem ; Weight</subject><ispartof>Journal of global optimization, 2019-07, Vol.74 (3), p.495-522</ispartof><rights>Springer Science+Business Media, LLC, part of Springer Nature 2019</rights><rights>COPYRIGHT 2019 Springer</rights><rights>Journal of Global Optimization is a copyright of Springer, (2019). All Rights Reserved.</rights><lds50>peer_reviewed</lds50><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c358t-ca7468516a44741218bd366562eb414b78dc0466d0cb3f2bb131078d970d16b3</citedby><cites>FETCH-LOGICAL-c358t-ca7468516a44741218bd366562eb414b78dc0466d0cb3f2bb131078d970d16b3</cites><orcidid>0000-0002-8528-3331</orcidid></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktopdf>$$Uhttps://link.springer.com/content/pdf/10.1007/s10898-019-00745-6$$EPDF$$P50$$Gspringer$$H</linktopdf><linktohtml>$$Uhttps://link.springer.com/10.1007/s10898-019-00745-6$$EHTML$$P50$$Gspringer$$H</linktohtml><link.rule.ids>314,780,784,27924,27925,41488,42557,51319</link.rule.ids></links><search><creatorcontrib>Schulze, Britta</creatorcontrib><creatorcontrib>Klamroth, Kathrin</creatorcontrib><creatorcontrib>Stiglmayr, Michael</creatorcontrib><title>Multi-objective unconstrained combinatorial optimization: a polynomial bound on the number of extreme supported solutions</title><title>Journal of global optimization</title><addtitle>J Glob Optim</addtitle><description>The multi-objective unconstrained combinatorial optimization problem (MUCO) can be considered as an archetype of a discrete linear multi-objective optimization problem. It can be interpreted as a specific relaxation of any multi-objective combinatorial optimization problem with linear sum objective function. While its single criteria analogon is analytically solvable, MUCO shares the computational complexity issues of most multi-objective combinatorial optimization problems: intractability and NP-hardness of the
ε
-constraint scalarizations. In this article interrelations between the supported points of a MUCO problem, arrangements of hyperplanes and a weight space decomposition, and zonotopes are presented. Based on these interrelations and a result by Zaslavsky on the number of faces in an arrangement of hyperplanes, a polynomial bound on the number of extreme supported solutions can be derived, leading to an exact polynomial time algorithm to find all extreme supported solutions. 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ε
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| subjects | Algorithms Combinatorial analysis Computer Science Extreme values Hyperplanes Mathematical programming Mathematics Mathematics and Statistics Mechanical properties Multiple objective analysis Operations Research/Decision Theory Optimization Polynomials Real Functions Traveling salesman problem Weight |
| title | Multi-objective unconstrained combinatorial optimization: a polynomial bound on the number of extreme supported solutions |
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