Hilbert basis (linear programming)

In linear programming, a Hilbert basis for a convex cone is a minimal set of integer vectors such that every integer vector in the convex cone is a linear combination of the vectors in the Hilbert basis with non-negative integer coefficients.

More precisely, a set $\{a\_1,ldots,a\_n\}$ of integer vectors is a Hilbert basis ifevery integer vector in its convex cone

:$\{\; lambda\_1\; a\_1\; +\; ldots\; +\; lambda\_n\; a\_n\; mid\; lambda\_1,ldots,lambda\_n\; geq\; 0,\; lambda\_1,ldots,lambda\_n\; mbox\{\; real\}\}$

is also in its integer cone

:$\{\; lambda\_1\; a\_1\; +\; ldots\; +\; lambda\_n\; a\_n\; mid\; lambda\_1,ldots,lambda\_n\; geq\; 0,\; lambda\_1,ldots,lambda\_n\; mbox\{\; integer\}\}.$

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