v2009.01.01 - Convex Optimization

2.7. CONES 87
2.6.1.3.1 Definition. Conventional boundary of convex set. [173,C.3.1]
The relative boundary ∂ C of a nonempty convex set C is the union of all the
exposed faces of C .
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Equivalence of this definition to (153) comes about because it is
conventionally presumed that any supporting hyperplane, central to the
definition of exposure, does not contain C . [266, p.100]
Any face F of convex set C (that is not C itself) belongs to rel∂C .
(2.8.2.1) In the exception when C is a single point {x} , (11)
rel∂{x} = {x}\{x} = ∅ , x∈ R n (154)
A bounded convex polyhedron (2.3.2,2.12.0.0.1) having nonempty interior,
for example, in R has a boundary constructed from two points, in R 2 from
at least three line segments, in R 3 from convex polygons, while a convex
polychoron (a bounded polyhedron in R 4 [326]) has a boundary constructed
from three-dimensional convex polyhedra. A halfspace is partially bounded
by a hyperplane; its interior therefore excludes that hyperplane. By
Definition 2.6.1.3.1, an affine set has no relative boundary.
2.7 Cones
In optimization, convex cones achieve prominence because they generalize
subspaces. Most compelling is the projection analogy: Projection on a
subspace can be ascertained from projection on its orthogonal complement
(E), whereas projection on a closed convex cone can be determined from
projection instead on its algebraic complement (2.13,E.9.2.1); called the
polar cone.
2.7.0.0.1 Definition. Ray.
The one-dimensional set
{ζΓ + B | ζ ≥ 0, Γ ≠ 0} ⊂ R n (155)
defines a halfline called a ray in nonzero direction Γ∈ R n having base
B ∈ R n . When B=0, a ray is the conic hull of direction Γ ; hence a
convex cone.
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