In linear programming

Linear programming

Linear programming is a mathematical method for determining a way to achieve the best outcome in a given mathematical model for some list of requirements represented as linear relationships...

, a Hilbert basis for a convex cone

Convex cone

In linear algebra, a convex cone is a subset of a vector space over an ordered field that is closed under linear combinations with positive coefficients.-Definition:...

 C is an integer cone basis: minimal set of integer vectors such that every integer vector in C is a conical combination

Conical combination

Given a finite number of vectors x_1, x_2, \dots, x_n\, in a real vector space, a conical combination or a conical sum of these vectors is a vector of the formwhere the real numbers \alpha_i\, satisfy \alpha_i\ge 0...

of the vectors in the Hilbert basis with integer coefficients.

Definition

A set of integer vectors is a Hilbert basis if every integer vector in its convex cone


is also in its integer cone