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| In [[linear programming]], a '''Hilbert basis''' for a [[convex cone]] ''C'' is an integer [[cone basis]]: minimal set of integer [[vector (mathematics)|vector]]s such that every integer vector in ''C'' is a [[conical combination]] of the vectors in the Hilbert basis with integer coefficients.
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| == Definition ==
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| A set <math>A=\{a_1,\ldots,a_n\}</math> of integer vectors is a Hilbert basis of its [[convex cone]]
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| :<math>C=\{ \lambda_1 a_1 + \ldots + \lambda_n a_n \mid \lambda_1,\ldots,\lambda_n \geq 0, \lambda_1,\ldots,\lambda_n \in\mathbb{R}\}</math>
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| if every integer vector from ''C'' belongs to the integer convex cone of ''A'':
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| :<math>\{ \alpha_1 a_1 + \ldots + \alpha_n a_n \mid \alpha_1,\ldots,\alpha_n \geq 0, \alpha_1,\ldots,\alpha_n \in\mathbb{Z}\},</math>
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| and no vector from ''A'' belongs to the integer convex cone of the others.
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| == References ==
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| * {{Citation | last1=Bruns | first1=Winfried |last2=Gubeladze | first2=Joseph | last3=Henk | first3=Martin | last4=Martin | first4=Alexander | last5=Weismantel | first5=Robert | title=A counterexample to an integer analogue of Carathéodory's theorem | doi=10.1515/crll.1999.045 | year=1999 | journal=[[Journal für die reine und angewandte Mathematik]] | volume=510 | pages=179–185 | issue=510}}
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| * {{Citation | last1=Cook | first1=William John | last2=Fonlupt | first2=Jean | last3=Schrijver | first3=Alexander | authorlink3=Alexander Schrijver|title=An integer analogue of Carathéodory's theorem | doi=10.1016/0095-8956(86)90064-X | year=1986 | journal=Journal of Combinatorial Theory. Series B | volume=40 | issue=1 | pages=63–70}}
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| * {{Citation | last1=Eisenbrand | first1=Friedrich | last2=Shmonin | first2=Gennady | title=Carathéodory bounds for integer cones | doi=10.1016/j.orl.2005.09.008 | year=2006 | journal=Operations Research Letters | volume=34 | issue=5 | pages=564–568}}
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| * {{cite journal |author = D. V. Pasechnik |title=On computing the Hilbert bases via the Elliott—MacMahon algorithm |journal=Theoretical Computer Science |volume=263 |year=2001 |pages=37–46 |doi=10.1016/S0304-3975(00)00229-2}}
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| {{mathapplied-stub}}
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| [[Category:Mathematical optimization]]
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| [[Category:Operations research]]
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| [[Category:Linear programming]]
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Hello. Let me introduce the author. Her title is Refugia Shryock. For a while I've been in South Dakota and my parents reside close by. The preferred pastime for my kids and me is to play baseball but I haven't made a dime with it. Hiring is her working day occupation now and she will not alter it whenever soon.
Feel free to visit my site; at home std test