Hilbert basis (linear programming)

The Hilbert basis of a convex cone C is a minimal set of integer vectors such that every integer vector in C is a conical combination of the vectors in the Hilbert basis with integer coefficients.

## Definition

Hilbert basis visualization

Given a lattice L⊂ℤd and a convex polyhedral cone with generators a1,…,an∈ℤd

C={λ1a1+…+λnan∣λ1,…,λn≥0,λ1,…,λn∈ℝ}⊂ℝd,

we consider the monoid C∩L. By Gordan's lemma, this monoid is finitely generated, i.e., there exists a finite set of lattice points {x1,…,xm}⊂C∩L such that every lattice point x∈C∩L is an integer conical combination of these points:

x=λ1x1+…+λmxm,λ1,…,λm∈ℤ,λ1,…,λm≥0.

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

## References

* Bruns, Winfried; Gubeladze, Joseph; Henk, Martin; Martin, Alexander; Weismantel, Robert (1999), "A counterexample to an integer analogue of Carathéodory's theorem", Journal für die reine und angewandte Mathematik 1999 (510): 179–185, doi:10.1515/crll.1999.045
* Cook, William John; Fonlupt, Jean; Schrijver, Alexander (1986), "An integer analogue of Carathéodory's theorem", Journal of Combinatorial Theory, Series B 40 (1): 63–70, doi:10.1016/0095-8956(86)90064-X
* Eisenbrand, Friedrich; Shmonin, Gennady (2006), "Carathéodory bounds for integer cones", Operations Research Letters 34 (5): 564–568, doi:10.1016/j.orl.2005.09.008, http://infoscience.epfl.ch/record/121640
* D. V. Pasechnik (2001). "On computing the Hilbert bases via the Elliott—MacMahon algorithm". Theoretical Computer Science 263 (1–2): 37–46. doi:10.1016/S0304-3975(00)00229-2.

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