Scanning integer points with lex-inequalities: a finite cutting plane algorithm for integer programming with linear objective
We consider the integer points in a unimodular cone K ordered by a lexicographic rule defined by a lattice basis. To each integer point x in K we associate a family of inequalities (lex-inequalities) that define the convex hull of the integer points in K that are not lexicographically smaller than x...
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| Veröffentlicht in: | 4OR 2021-12, Vol.19 (4), p.531-548 |
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| creator | Conforti, Michele De Santis, Marianna Di Summa, Marco Rinaldi, Francesco |
| description | We consider the integer points in a unimodular cone
K
ordered by a lexicographic rule defined by a lattice basis. To each integer point
x
in
K
we associate a family of inequalities (lex-inequalities) that define the convex hull of the integer points in
K
that are not lexicographically smaller than
x
. The family of lex-inequalities contains the Chvátal–Gomory cuts, but does not contain and is not contained in the family of split cuts. This provides a finite cutting plane method to solve the integer program
min
{
c
x
:
x
∈
S
∩
Z
n
}
, where
S
⊂
R
n
is a compact set and
c
∈
Z
n
. We analyze the number of iterations of our algorithm. |
| doi_str_mv | 10.1007/s10288-020-00459-6 |
| format | Article |
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K
ordered by a lexicographic rule defined by a lattice basis. To each integer point
x
in
K
we associate a family of inequalities (lex-inequalities) that define the convex hull of the integer points in
K
that are not lexicographically smaller than
x
. The family of lex-inequalities contains the Chvátal–Gomory cuts, but does not contain and is not contained in the family of split cuts. This provides a finite cutting plane method to solve the integer program
min
{
c
x
:
x
∈
S
∩
Z
n
}
, where
S
⊂
R
n
is a compact set and
c
∈
Z
n
. We analyze the number of iterations of our algorithm.</description><identifier>ISSN: 1619-4500</identifier><identifier>EISSN: 1614-2411</identifier><identifier>DOI: 10.1007/s10288-020-00459-6</identifier><language>eng</language><publisher>Berlin/Heidelberg: Springer Berlin Heidelberg</publisher><subject>Algorithms ; Business and Management ; Computational geometry ; Convexity ; Cutting ; Industrial and Production Engineering ; Inequalities ; Integer programming ; Linear programming ; Operations research ; Operations Research/Decision Theory ; Optimization ; Polytopes ; Research Paper</subject><ispartof>4OR, 2021-12, Vol.19 (4), p.531-548</ispartof><rights>The Author(s) 2020</rights><rights>The Author(s) 2020. 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-c453t-b724e51950dbc9640c0625e7a677b42532cc77693963af02d5d3c73fcfe19f6f3</citedby><cites>FETCH-LOGICAL-c453t-b724e51950dbc9640c0625e7a677b42532cc77693963af02d5d3c73fcfe19f6f3</cites><orcidid>0000-0002-1189-5917</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/s10288-020-00459-6$$EPDF$$P50$$Gspringer$$Hfree_for_read</linktopdf><linktohtml>$$Uhttps://link.springer.com/10.1007/s10288-020-00459-6$$EHTML$$P50$$Gspringer$$Hfree_for_read</linktohtml><link.rule.ids>314,776,780,27901,27902,41464,42533,51294</link.rule.ids></links><search><creatorcontrib>Conforti, Michele</creatorcontrib><creatorcontrib>De Santis, Marianna</creatorcontrib><creatorcontrib>Di Summa, Marco</creatorcontrib><creatorcontrib>Rinaldi, Francesco</creatorcontrib><title>Scanning integer points with lex-inequalities: a finite cutting plane algorithm for integer programming with linear objective</title><title>4OR</title><addtitle>4OR-Q J Oper Res</addtitle><description>We consider the integer points in a unimodular cone
K
ordered by a lexicographic rule defined by a lattice basis. To each integer point
x
in
K
we associate a family of inequalities (lex-inequalities) that define the convex hull of the integer points in
K
that are not lexicographically smaller than
x
. The family of lex-inequalities contains the Chvátal–Gomory cuts, but does not contain and is not contained in the family of split cuts. This provides a finite cutting plane method to solve the integer program
min
{
c
x
:
x
∈
S
∩
Z
n
}
, where
S
⊂
R
n
is a compact set and
c
∈
Z
n
. We analyze the number of iterations of our algorithm.</description><subject>Algorithms</subject><subject>Business and Management</subject><subject>Computational geometry</subject><subject>Convexity</subject><subject>Cutting</subject><subject>Industrial and Production Engineering</subject><subject>Inequalities</subject><subject>Integer programming</subject><subject>Linear programming</subject><subject>Operations research</subject><subject>Operations Research/Decision Theory</subject><subject>Optimization</subject><subject>Polytopes</subject><subject>Research Paper</subject><issn>1619-4500</issn><issn>1614-2411</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2021</creationdate><recordtype>article</recordtype><sourceid>C6C</sourceid><sourceid>BENPR</sourceid><recordid>eNp9kMtKAzEUhgdRsFZfwFXAdfTkMsmMOxFvUHChrkOaJjVlmmmT1MvCdzftCN25yoF8338Of1WdE7gkAPIqEaBNg4ECBuB1i8VBNSKCcEw5IYe7ucW8BjiuTlJaADDGOIyqnxejQ_BhjnzIdm4jWvVlSujT53fU2S_sg11vdOezt-kaaeR88Nkis8l5q606HSzS3byPxVgi18d9VOznUS-XW27IK2E6on66sCb7D3taHTndJXv2946rt_u719tHPHl-eLq9mWDDa5bxVFJua9LWMJuaVnAwIGhtpRZSTjmtGTVGStGyVjDtgM7qGTOSOeMsaZ1wbFxdDLnlovXGpqwW_SaGslLRhsimpVJCoehAmdinFK1Tq-iXOn4rAmpbsxpqVqVmtatZiSKhQbKmDz7tFdlQaKCRtCBsQFL5DKWY_fZ_gn8BoCmMlA</recordid><startdate>20211201</startdate><enddate>20211201</enddate><creator>Conforti, Michele</creator><creator>De Santis, Marianna</creator><creator>Di Summa, Marco</creator><creator>Rinaldi, Francesco</creator><general>Springer Berlin Heidelberg</general><general>Springer Nature B.V</general><scope>C6C</scope><scope>OQ6</scope><scope>AAYXX</scope><scope>CITATION</scope><scope>3V.</scope><scope>7WY</scope><scope>7WZ</scope><scope>7XB</scope><scope>87Z</scope><scope>8AL</scope><scope>8FE</scope><scope>8FG</scope><scope>8FK</scope><scope>8FL</scope><scope>ABJCF</scope><scope>ABUWG</scope><scope>AFKRA</scope><scope>ARAPS</scope><scope>AZQEC</scope><scope>BENPR</scope><scope>BEZIV</scope><scope>BGLVJ</scope><scope>CCPQU</scope><scope>DWQXO</scope><scope>FRNLG</scope><scope>F~G</scope><scope>GNUQQ</scope><scope>HCIFZ</scope><scope>JQ2</scope><scope>K60</scope><scope>K6~</scope><scope>K7-</scope><scope>L.-</scope><scope>L6V</scope><scope>M0C</scope><scope>M0N</scope><scope>M7S</scope><scope>P62</scope><scope>PQBIZ</scope><scope>PQBZA</scope><scope>PQEST</scope><scope>PQQKQ</scope><scope>PQUKI</scope><scope>PTHSS</scope><scope>PYYUZ</scope><scope>Q9U</scope><orcidid>https://orcid.org/0000-0002-1189-5917</orcidid></search><sort><creationdate>20211201</creationdate><title>Scanning integer points with lex-inequalities: a finite cutting plane algorithm for integer programming with linear objective</title><author>Conforti, Michele ; De Santis, Marianna ; Di Summa, Marco ; Rinaldi, Francesco</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c453t-b724e51950dbc9640c0625e7a677b42532cc77693963af02d5d3c73fcfe19f6f3</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2021</creationdate><topic>Algorithms</topic><topic>Business and Management</topic><topic>Computational geometry</topic><topic>Convexity</topic><topic>Cutting</topic><topic>Industrial and Production Engineering</topic><topic>Inequalities</topic><topic>Integer programming</topic><topic>Linear programming</topic><topic>Operations research</topic><topic>Operations Research/Decision Theory</topic><topic>Optimization</topic><topic>Polytopes</topic><topic>Research Paper</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Conforti, Michele</creatorcontrib><creatorcontrib>De Santis, Marianna</creatorcontrib><creatorcontrib>Di Summa, Marco</creatorcontrib><creatorcontrib>Rinaldi, Francesco</creatorcontrib><collection>Springer Nature OA Free Journals</collection><collection>ECONIS</collection><collection>CrossRef</collection><collection>ProQuest Central (Corporate)</collection><collection>ABI/INFORM Collection</collection><collection>ABI/INFORM Global (PDF only)</collection><collection>ProQuest Central (purchase pre-March 2016)</collection><collection>ABI/INFORM Global (Alumni Edition)</collection><collection>Computing Database (Alumni Edition)</collection><collection>ProQuest SciTech Collection</collection><collection>ProQuest Technology Collection</collection><collection>ProQuest Central (Alumni) (purchase pre-March 2016)</collection><collection>ABI/INFORM Collection (Alumni Edition)</collection><collection>Materials Science & Engineering Collection</collection><collection>ProQuest Central (Alumni Edition)</collection><collection>ProQuest Central UK/Ireland</collection><collection>Advanced Technologies & Aerospace Collection</collection><collection>ProQuest Central Essentials</collection><collection>ProQuest Central</collection><collection>Business Premium Collection</collection><collection>Technology Collection</collection><collection>ProQuest One Community College</collection><collection>ProQuest Central Korea</collection><collection>Business Premium Collection (Alumni)</collection><collection>ABI/INFORM Global (Corporate)</collection><collection>ProQuest Central Student</collection><collection>SciTech Premium Collection</collection><collection>ProQuest Computer Science Collection</collection><collection>ProQuest Business Collection (Alumni Edition)</collection><collection>ProQuest Business Collection</collection><collection>Computer Science Database</collection><collection>ABI/INFORM Professional Advanced</collection><collection>ProQuest Engineering Collection</collection><collection>ABI/INFORM Global</collection><collection>Computing Database</collection><collection>Engineering Database</collection><collection>ProQuest Advanced Technologies & Aerospace Collection</collection><collection>ProQuest One Business</collection><collection>ProQuest One Business (Alumni)</collection><collection>ProQuest One Academic Eastern Edition (DO NOT USE)</collection><collection>ProQuest One Academic</collection><collection>ProQuest One Academic UKI Edition</collection><collection>Engineering Collection</collection><collection>ABI/INFORM Collection China</collection><collection>ProQuest Central Basic</collection><jtitle>4OR</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Conforti, Michele</au><au>De Santis, Marianna</au><au>Di Summa, Marco</au><au>Rinaldi, Francesco</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Scanning integer points with lex-inequalities: a finite cutting plane algorithm for integer programming with linear objective</atitle><jtitle>4OR</jtitle><stitle>4OR-Q J Oper Res</stitle><date>2021-12-01</date><risdate>2021</risdate><volume>19</volume><issue>4</issue><spage>531</spage><epage>548</epage><pages>531-548</pages><issn>1619-4500</issn><eissn>1614-2411</eissn><abstract>We consider the integer points in a unimodular cone
K
ordered by a lexicographic rule defined by a lattice basis. To each integer point
x
in
K
we associate a family of inequalities (lex-inequalities) that define the convex hull of the integer points in
K
that are not lexicographically smaller than
x
. The family of lex-inequalities contains the Chvátal–Gomory cuts, but does not contain and is not contained in the family of split cuts. This provides a finite cutting plane method to solve the integer program
min
{
c
x
:
x
∈
S
∩
Z
n
}
, where
S
⊂
R
n
is a compact set and
c
∈
Z
n
. We analyze the number of iterations of our algorithm.</abstract><cop>Berlin/Heidelberg</cop><pub>Springer Berlin Heidelberg</pub><doi>10.1007/s10288-020-00459-6</doi><tpages>18</tpages><orcidid>https://orcid.org/0000-0002-1189-5917</orcidid><oa>free_for_read</oa></addata></record> |
| fulltext | fulltext |
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| issn | 1619-4500 1614-2411 |
| language | eng |
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| source | SpringerLink Journals |
| subjects | Algorithms Business and Management Computational geometry Convexity Cutting Industrial and Production Engineering Inequalities Integer programming Linear programming Operations research Operations Research/Decision Theory Optimization Polytopes Research Paper |
| title | Scanning integer points with lex-inequalities: a finite cutting plane algorithm for integer programming with linear objective |
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