Hilbert basis (linear programming)

In linear programming, a Hilbert basis for a convex cone C is an integer cone basis: 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

A set $A=\{a_{1},\ldots ,a_{n}\}$ of integer vectors is a Hilbert basis of its convex cone

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}}\}

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

\{\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}}\},

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

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