v2010.10.26 - Convex Optimization

2.6. EXTREME, EXPOSED 93A one-dimensional face of a convex set is called an edge.△Dimension of a face is the penultimate number of affinely independentpoints (2.4.2.3) belonging to it;dim F = sup dim{x 2 − x 1 , x 3 − x 1 , ... , x ρ − x 1 | x i ∈ F , i=1... ρ} (169)ρThe point of intersection in C with a strictly supporting hyperplaneidentifies an extreme point, but not vice versa. The nonempty intersection ofany supporting hyperplane with C identifies a face, in general, but not viceversa. To acquire a converse, the concept exposed face requires introduction:2.6.1 Exposure2.6.1.0.1 Definition. Exposed face, exposed point, vertex, facet.[199,A.2.3, A.2.4]Fis an exposed face of an n-dimensional convex set C iff there is asupporting hyperplane ∂H to C such thatF = C ∩ ∂H (170)Only faces of dimension −1 through n −1 can be exposed by ahyperplane.An exposed point, the definition of vertex, is equivalent to azero-dimensional exposed face; the point of intersection with a strictlysupporting hyperplane.Afacet is an (n −1)-dimensional exposed face of an n-dimensionalconvex set C ; facets exist in one-to-one correspondence with the(n −1)-dimensional faces. 2.26{exposed points} = {extreme points}{exposed faces} ⊆ {faces}△2.26 This coincidence occurs simply because the facet’s dimension is the same as thedimension of the supporting hyperplane exposing it.