v2009.01.01 - Convex Optimization

2.7. CONES 91
2.7.2 Convex cone
We call the set K ⊆ R M a convex cone iff
Γ 1 , Γ 2 ∈ K ⇒ ζΓ 1 + ξΓ 2 ∈ K for all ζ,ξ ≥ 0 (157)
Apparent from this definition, ζΓ 1 ∈ K and ξΓ 2 ∈ K for all ζ,ξ ≥ 0. The
set K is convex since, for any particular ζ,ξ ≥ 0
because µζ,(1 − µ)ξ ≥ 0.
Obviously,
µζΓ 1 + (1 − µ)ξΓ 2 ∈ K ∀µ ∈ [0, 1] (158)
{X } ⊃ {K} (159)
the set of all convex cones is a proper subset of all cones. The set of
convex cones is a narrower but more familiar class of cone, any member
of which can be equivalently described as the intersection of a possibly
(but not necessarily) infinite number of hyperplanes (through the origin)
and halfspaces whose bounding hyperplanes pass through the origin; a
halfspace-description (2.4). Convex cones need not be full-dimensional.
Figure 33: Not a cone; ironically, the three-dimensional flared horn (with or
without its interior) resembling the mathematical symbol ≻ denoting cone
membership and partial order.