v2007.09.13 - Convex Optimization

2.8. CONE BOUNDARY 91Proper cone {0} in R 0 has no boundary (140) because (11)rel int{0} = {0} (154)The boundary of any proper cone in R is the origin.The boundary of any convex cone whose dimension exceeds 1 can beconstructed entirely from an aggregate of rays emanating exclusively fromthe origin.2.8.1 Extreme directionThe property extreme direction arises naturally in connection with thepointed closed convex cone K ⊂ R n , being analogous to extreme point.[228,18, p.162] 2.25 An extreme direction Γ ε of pointed K is a vectorcorresponding to an edge that is a ray emanating from the origin. 2.26Nonzero direction Γ ε in pointed K is extreme if and only ifζ 1 Γ 1 +ζ 2 Γ 2 ≠ Γ ε ∀ ζ 1 , ζ 2 ≥ 0, ∀ Γ 1 , Γ 2 ∈ K\{ζΓ ε ∈ K | ζ ≥0} (155)In words, an extreme direction in a pointed closed convex cone is thedirection of a ray, called an extreme ray, that cannot be expressed as a coniccombination of any ray directions in the cone distinct from it.An extreme ray is a one-dimensional face of K . By (85), extremedirection Γ ε is not a point relatively interior to any line segment inK\{ζΓ ε ∈ K | ζ ≥0}. Thus, by analogy, the corresponding extreme ray{ζΓ ε ∈ K | ζ ≥0} is not a ray relatively interior to any plane segment 2.27in K .2.8.1.1 extreme distinction, uniquenessAn extreme direction is unique, but its vector representation Γ ε is notbecause any positive scaling of it produces another vector in the same(extreme) direction. Hence an extreme direction is unique to within a positivescaling. When we say extreme directions are distinct, we are referring todistinctness of rays containing them. Nonzero vectors of various length in2.25 We diverge from Rockafellar’s extreme direction: “extreme point at infinity”.2.26 An edge (2.6.0.0.3) of a convex cone is not necessarily a ray. A convex cone maycontain an edge that is a line; e.g., a wedge-shaped polyhedral cone (K ∗ in Figure 29).2.27 A planar fragment; in this context, a planar cone.