v2010.10.26 - Convex Optimization

2.8. CONE BOUNDARY 107That means there exists a supporting hyperplane ∂H to K containing 0Γ .So the ray through Γ belongs both to K and to ∂H . ∂H must thereforeexpose a face of K that contains the ray; id est,{ζΓ | ζ ≥ 0} ⊆ K ∩ ∂H ⊂ ∂K (184)Proper cone {0} in R 0 has no boundary (24) because (12)rel int{0} = {0} (185)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.[307,18, p.162] 2.33 An extreme direction Γ ε of pointed K is a vectorcorresponding to an edge that is a ray emanating from the origin. 2.34Nonzero 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} (186)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 directions of any rays in the cone distinct from it.An extreme ray is a one-dimensional face of K . By (105), 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 2.35in K .2.33 We diverge from Rockafellar’s extreme direction: “extreme point at infinity”.2.34 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 41).2.35 A planar fragment; in this context, a planar cone.