v2009.01.01 - Convex Optimization

2.12. CONVEX POLYHEDRA 133
It follows from definition (258) by exposure that each face of a convex
polyhedron is a convex polyhedron.
The projection of any polyhedron on a subspace remains a polyhedron.
More generally, the image of a polyhedron under any linear transformation
is a polyhedron. [25,I.9]
When b and d in (258) are 0, the resultant is a polyhedral cone. The
set of all polyhedral cones is a subset of convex cones:
2.12.1 Polyhedral cone
From our study of cones, we see: the number of intersecting hyperplanes and
halfspaces constituting a convex cone is possibly but not necessarily infinite.
When the number is finite, the convex cone is termed polyhedral. That is
the primary distinguishing feature between the set of all convex cones and
polyhedra; all polyhedra, including polyhedral cones, are finitely generated
[266,19]. We distinguish polyhedral cones in the set of all convex cones for
this reason, although all convex cones of dimension 2 or less are polyhedral.
2.12.1.0.1 Definition. Polyhedral cone, halfspace-description. 2.46
(confer (94)) A polyhedral cone is the intersection of a finite number of
halfspaces and hyperplanes about the origin;
K = {y | Ay ≽ 0, Cy = 0} ⊆ R n (a)
= {y | Ay ≽ 0, Cy ≽ 0, Cy ≼ 0} (b)
⎧ ⎡ ⎤ ⎫
⎨ A ⎬
=
⎩ y | ⎣ C ⎦y ≽ 0
(c)
⎭
−C
(259)
where coefficients A and C generally denote matrices of finite dimension.
Each row of C is a vector normal to a hyperplane containing the origin,
while each row of A is a vector inward-normal to a hyperplane containing
the origin and partially bounding a halfspace.
△
A polyhedral cone thus defined is closed, convex, possibly has empty
interior, and only a finite number of generators (2.8.1.2), and vice versa.
(Minkowski/Weyl) [286,2.8] [266, thm.19.1]
2.46 Rockafellar [266,19] proposes affine sets be handled via complementary pairs of affine
inequalities; e.g., Cy ≽d and Cy ≼d .