v2007.11.26 - Convex Optimization

2.7. CONES 812.6.1.3.1 Definition. Conventional boundary of convex set. [148,C.3.1]The relative boundary ∂ C of a nonempty convex set C is the union of all theexposed faces of C .△Equivalence of this definition to (141) comes about because it isconventionally presumed that any supporting hyperplane, central to thedefinition of exposure, does not contain C . [232, p.100]Any face F of convex set C (that is not C itself) belongs to rel∂C .(2.8.2.1) In the exception when C is a single point {x} , (11)rel∂{x} = {x}\{x} = ∅ , x∈ R n (142)A bounded convex polyhedron (2.12.0.0.1) having nonempty interior, forexample, in R has a boundary constructed from two points, in R 2 fromat least three line segments, in R 3 from convex polygons, while a convexpolychoron (a bounded polyhedron in R 4 [284]) has a boundary constructedfrom three-dimensional convex polyhedra.By Definition 2.6.1.3.1, an affine set has no relative boundary.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(E), whereas projection on a closed convex cone can be determined fromprojection instead on its algebraic complement (2.13,E.9.2.1); called thepolar cone.2.7.0.0.1 Definition. Ray.The one-dimensional set{ζΓ + B | ζ ≥ 0, Γ ≠ 0} ⊂ R n (143)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 a convexcone.△The conventional boundary of a single ray, base 0, in any dimension isthe origin because that is the union of all exposed faces not containing theentire set. Its relative interior is the ray itself excluding the origin.