v2004.06.19 - Convex Optimization

2.5. EXTREME, EXPOSED 49• A facet is an (n − 1)-dimensional exposed face of an n-dimensionalconvex set C ; in one-to-one correspondence with the (n−1)-dimensionalfaces. 2.16• {exposed points} = {extreme points}{exposed faces} ⊆ {faces}△2.5.1.1 Density of exposed pointsFor any closed convex set C , its exposed points constitute a dense subset ofits extreme points; [30, §18] [56] [31, §3.6, p.115] dense in the sense [48] thatclosure of that subset yields the set of extreme points.For the convex set illustrated in Figure 2.7, 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 C .2.5.1.2 Face transitivity and algebraFaces enjoy a transitive relationship. If F 1 is a face (an extreme set) of F 2which, in turn, is a face of F 3 , then it is always true that F 1 is a face ofF 3 . [30, §18] [57, def.115/6, p.358] For example, any extreme point of F 2 isan extreme point of F 3 . (The parallel statement for exposed faces is false.)Yet it is erroneous to presume that a face, of dimension 1 or more, consistsentirely of extreme points, nor is a face of dimension 2 or more entirelycomposed of edges, and so on.For the polyhedron in R 3 from Figure 2.2, for example, the nonemptyfaces exposed by a hyperplane are the vertices, edges, and facets; there areno more. The zero-, one-, and two-dimensional faces are in one-to-one correspondencewith the exposed faces in that example.Define the smallest face F of a convex set C containing some element G :F(C ∋G) (104)2.16 This coincidence occurs simply because the facet’s dimension is the same as the dimensionof the supporting hyperplane exposing it.