Approximating biobjective minimization problems using general ordering cones
Abstract This article investigates the approximation quality achievable for biobjective minimization problems with respect to the Pareto cone by solutions that are (approximately) optimal with respect to larger ordering cones. When simultaneously considering α-approximations for all closed convex or...
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| Veröffentlicht in: | Journal of global optimization 2023-06, Vol.86 (2), p.393-415 |
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| container_title | Journal of global optimization |
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| creator | Herzel, Arne Helfrich, Stephan Ruzika, Stefan Thielen, Clemens |
| description | Abstract
This article investigates the approximation quality achievable for biobjective minimization problems with respect to the Pareto cone by solutions that are (approximately) optimal with respect to larger ordering cones. When simultaneously considering α-approximations for all closed convex ordering cones of a fixed inner angle γ∈π2,π, an approximation guarantee between αand 2αis achieved, which depends continuously on γ. The analysis is best-possible for any inner angle and it generalizes and unifies the known results that the set of supported solutions is a 2-approximation and that the efficient set itself is a 1-approximation. Moreover, it is shown that, for maximization problems, no approximation guarantee is achievable in general by considering larger ordering cones in the described fashion, which again generalizes a known result about the set of supported solutions. |
| doi_str_mv | 10.1007/s10898-023-01276-x |
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This article investigates the approximation quality achievable for biobjective minimization problems with respect to the Pareto cone by solutions that are (approximately) optimal with respect to larger ordering cones. When simultaneously considering α-approximations for all closed convex ordering cones of a fixed inner angle γ∈π2,π, an approximation guarantee between αand 2αis achieved, which depends continuously on γ. The analysis is best-possible for any inner angle and it generalizes and unifies the known results that the set of supported solutions is a 2-approximation and that the efficient set itself is a 1-approximation. Moreover, it is shown that, for maximization problems, no approximation guarantee is achievable in general by considering larger ordering cones in the described fashion, which again generalizes a known result about the set of supported solutions.</description><identifier>ISSN: 1573-2916</identifier><identifier>ISSN: 0925-5001</identifier><identifier>EISSN: 1573-2916</identifier><identifier>DOI: 10.1007/s10898-023-01276-x</identifier><language>eng</language><publisher>New York: Springer US</publisher><subject>Approximate Pareto set ; Approximation ; Computer Science ; Cones ; Efficient solution ; Guarantees ; Mathematical analysis ; Mathematics ; Mathematics and Statistics ; Multiobjective optimization ; Operations Research/Decision Theory ; Optimization ; Ordering cone ; Preferences ; Real Functions ; Supported solution</subject><ispartof>Journal of global optimization, 2023-06, Vol.86 (2), p.393-415</ispartof><rights>The Author(s) 2023</rights><rights>COPYRIGHT 2023 Springer</rights><rights>The Author(s) 2023. This work is published under http://creativecommons.org/licenses/by/4.0/ (the “License”). Notwithstanding the ProQuest Terms and Conditions, you may use this content in accordance with the terms of the License.</rights><lds50>peer_reviewed</lds50><oa>free_for_read</oa><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c424t-4bd4b4f24921d0012f2e8e0419424b1f6db110d2946964ad0b11c4892810e8f3</citedby><cites>FETCH-LOGICAL-c424t-4bd4b4f24921d0012f2e8e0419424b1f6db110d2946964ad0b11c4892810e8f3</cites><orcidid>0000-0003-0897-3571</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/s10898-023-01276-x$$EPDF$$P50$$Gspringer$$Hfree_for_read</linktopdf><linktohtml>$$Uhttps://link.springer.com/10.1007/s10898-023-01276-x$$EHTML$$P50$$Gspringer$$Hfree_for_read</linktohtml><link.rule.ids>314,776,780,27903,27904,41467,42536,51297</link.rule.ids></links><search><creatorcontrib>Herzel, Arne</creatorcontrib><creatorcontrib>Helfrich, Stephan</creatorcontrib><creatorcontrib>Ruzika, Stefan</creatorcontrib><creatorcontrib>Thielen, Clemens</creatorcontrib><title>Approximating biobjective minimization problems using general ordering cones</title><title>Journal of global optimization</title><addtitle>J Glob Optim</addtitle><description>Abstract
This article investigates the approximation quality achievable for biobjective minimization problems with respect to the Pareto cone by solutions that are (approximately) optimal with respect to larger ordering cones. When simultaneously considering α-approximations for all closed convex ordering cones of a fixed inner angle γ∈π2,π, an approximation guarantee between αand 2αis achieved, which depends continuously on γ. The analysis is best-possible for any inner angle and it generalizes and unifies the known results that the set of supported solutions is a 2-approximation and that the efficient set itself is a 1-approximation. 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This article investigates the approximation quality achievable for biobjective minimization problems with respect to the Pareto cone by solutions that are (approximately) optimal with respect to larger ordering cones. When simultaneously considering α-approximations for all closed convex ordering cones of a fixed inner angle γ∈π2,π, an approximation guarantee between αand 2αis achieved, which depends continuously on γ. The analysis is best-possible for any inner angle and it generalizes and unifies the known results that the set of supported solutions is a 2-approximation and that the efficient set itself is a 1-approximation. Moreover, it is shown that, for maximization problems, no approximation guarantee is achievable in general by considering larger ordering cones in the described fashion, which again generalizes a known result about the set of supported solutions.</abstract><cop>New York</cop><pub>Springer US</pub><doi>10.1007/s10898-023-01276-x</doi><tpages>23</tpages><orcidid>https://orcid.org/0000-0003-0897-3571</orcidid><oa>free_for_read</oa></addata></record> |
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| subjects | Approximate Pareto set Approximation Computer Science Cones Efficient solution Guarantees Mathematical analysis Mathematics Mathematics and Statistics Multiobjective optimization Operations Research/Decision Theory Optimization Ordering cone Preferences Real Functions Supported solution |
| title | Approximating biobjective minimization problems using general ordering cones |
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