v2009.01.01 - Convex Optimization

2.8. CONE BOUNDARY 97
So the ray through Γ belongs both to K and to ∂H .
expose a face of K that contains the ray; id est,
∂H must therefore
{ζΓ | ζ ≥ 0} ⊆ K ∩ ∂H ⊂ ∂K (166)
Proper cone {0} in R 0 has no boundary (153) because (11)
rel int{0} = {0} (167)
The boundary of any proper cone in R is the origin.
The boundary of any convex cone whose dimension exceeds 1 can be
constructed entirely from an aggregate of rays emanating exclusively from
the origin.
2.8.1 Extreme direction
The property extreme direction arises naturally in connection with the
pointed closed convex cone K ⊂ R n , being analogous to extreme point.
[266,18, p.162] 2.31 An extreme direction Γ ε of pointed K is a vector
corresponding to an edge that is a ray emanating from the origin. 2.32
Nonzero direction Γ ε in pointed K is extreme if and only if
ζ 1 Γ 1 +ζ 2 Γ 2 ≠ Γ ε ∀ ζ 1 , ζ 2 ≥ 0, ∀ Γ 1 , Γ 2 ∈ K\{ζΓ ε ∈ K | ζ ≥0} (168)
In words, an extreme direction in a pointed closed convex cone is the
direction of a ray, called an extreme ray, that cannot be expressed as a conic
combination of directions of any rays in the cone distinct from it.
An extreme ray is a one-dimensional face of K . By (96), extreme
direction Γ ε is not a point relatively interior to any line segment in
K\{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 2.33
in K .
2.31 We diverge from Rockafellar’s extreme direction: “extreme point at infinity”.
2.32 An edge (2.6.0.0.3) of a convex cone is not necessarily a ray. A convex cone may
contain an edge that is a line; e.g., a wedge-shaped polyhedral cone (K ∗ in Figure 35).
2.33 A planar fragment; in this context, a planar cone.