v2007.09.13 - Convex Optimization

2.7. CONES 852.7.2 Convex coneWe call the set K ⊆ R M a convex cone iffΓ 1 , Γ 2 ∈ K ⇒ ζΓ 1 + ξΓ 2 ∈ K for all ζ,ξ ≥ 0 (144)Apparent from this definition, ζΓ 1 ∈ K and ξΓ 2 ∈ K for all ζ,ξ ≥ 0. Theset K is convex since, for any particular ζ,ξ ≥ 0because µ ζ,(1 − µ)ξ ≥ 0.Obviously,µ ζΓ 1 + (1 − µ)ξΓ 2 ∈ K ∀µ ∈ [0, 1] (145){X } ⊃ {K} (146)the set of all convex cones is a proper subset of all cones. The set ofconvex cones is a narrower but more familiar class of cone, any memberof 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; ahalfspace-description (2.4). The interior of a convex cone is possibly empty.Figure 27: Not a cone; ironically, the three-dimensional flared horn (with orwithout its interior) resembling the mathematical symbol ≻ denoting conemembership and partial order.