Читать книгу Optimizations and Programming - Bouchaib Radi, Ghias Kharmanda, Michel Ledoux - Страница 15

DEFINITION 1.2.– A polyhedron is a set that can be described as

where A is an m × n matrix and b is a vector in ℝ^m. Note that the admissible set of an LP is a polyhedron.

DEFINITION 1.3.– A polyhedron P is said to be bounded if there exists a constant c such that ||x|| ≤ c for every x ∈ P.

Figure 1.1. Bounded polyhedron (left) and unbounded polyhedron (right). For a color version of this figure, see www.iste.co.uk/radi/optimizations.zip

DEFINITION 1.4.– Let a be a non-zero vector of ℝⁿ, and let b be a scalar.

 – The set {x ∈ ℝn|aTx = b} is said to be a hyperplane.

 – The set {x ∈ ℝn|aTx ≥ b} is said to be a half-space.

Note that a hyperplane is the boundary of the corresponding half-space and the vector a is perpendicular to the hyperplane that it defines [TEG 12].