v2007.09.17 - Convex Optimization

2.6. EXTREME, EXPOSED 792.6.1 Exposure2.6.1.0.1 Definition. Exposed face, exposed point, vertex, facet.[148,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 (137)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 ; in one-to-one correspondence with the(n −1)-dimensional faces. 2.22{exposed points} = {extreme points}{exposed faces} ⊆ {faces}△2.6.1.1 Density of exposed pointsFor any closed convex set C , its exposed points constitute a dense subset ofits extreme points; [230,18] [252] [247,3.6, p.115] dense in the sense [282]that closure of that subset yields the set of extreme points.For the convex set illustrated in Figure 21, point B cannot be exposedbecause it relatively bounds both the facet AB and the closed quarter circle,each bounding the set. Since B is not relatively interior to any line segmentin the set, then B is an extreme point by definition. Point B may be regardedas the limit of some sequence of exposed points beginning at vertex C .2.22 This coincidence occurs simply because the facet’s dimension is the same as thedimension of the supporting hyperplane exposing it.