v2010.10.26 - Convex Optimization

96 CHAPTER 2. CONVEX GEOMETRY2.6.1.4 Conventional boundary(confer2.1.7.2) Relative boundaryis equivalent to:rel ∂ C = C \ rel int C (24)2.6.1.4.1 Definition. Conventional boundary of convex set. [199,C.3.1]The relative boundary ∂ C of a nonempty convex set C is the union of allexposed faces of C .△Equivalence to (24) comes about because it is conventionally presumedthat any supporting hyperplane, central to the definition of exposure, doesnot contain C . [307, p.100] Any face F of convex set C (that is not C itself)belongs to rel∂C . (2.8.2.1)2.7 ConesIn optimization, convex cones achieve prominence because they generalizesubspaces. Most compelling is the projection analogy: Projection on asubspace can be ascertained from projection on its orthogonal complement(Figure 165), whereas projection on a closed convex cone can be determinedfrom projection instead on its algebraic complement (2.13,E.9.2.1); calledthe polar cone.2.7.0.0.1 Definition. Ray.The one-dimensional set{ζΓ + B | ζ ≥ 0, Γ ≠ 0} ⊂ R n (173)defines a halfline called a ray in nonzero direction Γ∈ R n having baseB ∈ R n . When B=0, a ray is the conic hull of direction Γ ; hence aconvex cone.△Relative boundary of a single ray, base 0 in any dimension, is the originbecause that is the union of all exposed faces not containing the entire set.Its relative interior is the ray itself excluding the origin.