Hilbert basis (linear programming): Difference between revisions

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In [[linear programming]], aThe '''Hilbert basis''' forof a [[convex cone]] ''C'' is ana minimal set of integer [[conevector basis(mathematics)|vector]]:s minimalin set of integer vectors''C'' such that every [[integer]] vector in ''C'' is a [[conical combination]] of the vectors in the Hilbert basis with integer coefficients.

== Definition ==

[[File:Hilbert basis.gif|thumb|Hilbert basis visualization. Two rays in the plane define an infinite cone of all the points lying between them. The unique Hilbert basis points of the cone are circled in yellow. Every integer point in the cone can be written as a sum of these basis elements. As you change the cone by moving one of the rays, the Hilbert basis also changes.]]

A set <math>A=\{a_1,\ldots,a_n\}</math> of integer vectors is a Hilbert basis of its [[convex cone]]

Given a [[Lattice (group)|lattice]] <math>L\subset\mathbb{Z}^d</math> and a convex polyhedral cone with generators <math>a_1,\ldots,a_n\in\mathbb{Z}^d</math>

:<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}\}\subset\mathbb{R}^d,</math>

we consider the [[monoid]] <math>C\cap L</math>. By [[Gordan's lemma]], this monoid is finitely generated, i.e., there exists a [[finite set]] of lattice points <math>\{x_1,\ldots,x_m\}\subset C\cap L</math> such that every lattice point <math>x\in C\cap L</math> is an integer conical combination of these points:

if every integer vector from ''C'' belongs to the integer convex cone of ''A'':

:<math>\{ x=\alpha_1 a_1lambda_1 x_1+ \ldots + \alpha_nlambda_m a_nx_m, \mid quad\alpha_1lambda_1,\ldots,\alpha_n lambda_m\geq 0in\mathbb{Z}, \alpha_1lambda_1,\ldots,\alpha_n lambda_m\in\mathbb{Z}\},geq0.</math>

The cone ''C'' is called pointed if <math>x,-x\in C</math> implies <math>x=0</math>. In this case there exists a unique minimal generating set of the monoid <math>C\cap L</math>—the '''Hilbert basis''' of ''C''. It is given by the set of irreducible lattice points: An element <math>x\in C\cap L</math> is called irreducible if it can not be written as the sum of two non-zero elements, i.e., <math>x=y+z</math> implies <math>y=0</math> or <math>z=0</math>.

and no vector from ''A'' belongs to the integer convex cone of the others.

== References ==

* {{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=5101999 | pages=179–185 | issue=510}}

* {{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| doi-access=free }}

* {{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| url=http://infoscience.epfl.ch/record/121640 }}

* {{cite journal |author = D. V. Pasechnik |title=On computing the Hilbert bases via the Elliott—MacMahon algorithm |journal=Theoretical Computer Science |volume=263 |issue=1–2 |year=2001 |pages=37–46 |doi=10.1016/S0304-3975(00)00229-2|doi-access=free |hdl=10220/8240 |hdl-access=free }}

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