v2010.10.26 - Convex Optimization

2.6. EXTREME, EXPOSED 952.6.1.1 Density of exposed pointsFor any closed convex set C , its exposed points constitute a dense subset ofits extreme points; [307,18] [335] [330,3.6, p.115] dense in the sense [373]that closure of that subset yields the set of extreme points.For the convex set illustrated in Figure 32, 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 .2.6.1.2 Face transitivity and algebraFaces of a convex set enjoy transitive relation. If F 1 is a face (an extreme set)of F 2 which in turn is a face of F 3 , then it is always true that F 1 is aface of F 3 . (The parallel statement for exposed faces is false. [307,18])For example, any extreme point of F 2 is an extreme point of F 3 ; inthis example, F 2 could be a face exposed by a hyperplane supportingpolyhedron F 3 . [223, def.115/6 p.358] Yet it is erroneous to presume thata face, of dimension 1 or more, consists entirely of extreme points. Nor is aface of dimension 2 or more entirely composed of edges, and so on.For the polyhedron in R 3 from Figure 20, for example, the nonemptyfaces exposed by a hyperplane are the vertices, edges, and facets; thereare no more. The zero-, one-, and two-dimensional faces are in one-to-onecorrespondence with the exposed faces in that example.2.6.1.3 Smallest faceDefine the smallest face F , that contains some element G , of a convex set C :F(C ∋G) (171)videlicet, C ⊃ rel int F(C ∋G) ∋ G . An affine set has no faces except itselfand the empty set. The smallest face, that contains G , of the intersectionof convex set C with an affine set A [239,2.4] [240]F((C ∩ A)∋G) = F(C ∋G) ∩ A (172)equals the intersection of A with the smallest face, that contains G , of set C .