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_1,\ldots,a_n\} of integer vectors is a Hilbert basis if every integer vector in its convex cone

\{ \lambda_1 a_1 %2B \ldots %2B \lambda_n a_n \mid \lambda_1,\ldots,\lambda_n \geq 0, \lambda_1,\ldots,\lambda_n \in\mathbb{R}\}

is also in its integer cone

\{ \alpha_1 a_1 %2B \ldots %2B \alpha_n a_n \mid \alpha_1,\ldots,\alpha_n \geq 0, \alpha_1,\ldots,\alpha_n \in\mathbb{Z}\}.

References