v2006.03.09 - Convex Optimization

108 CHAPTER 2. CONVEX GEOMETRYFrom Theorem 2.8.1.1.1,rel∂C = C \ rel int C (134)= conv{exposed points and exposed rays} \ rel int C= conv{extreme points and extreme rays} \ rel int C⎫⎪⎬⎪⎭(153)Thus each and every extreme point of a convex set (that is not a point)resides on its relative boundary, while each and every extreme direction of aconvex set (that is not a halfline and contains no line) resides on its relativeboundary because extreme points and directions of such respective sets donot belong to the relative interior by definition.The relationship between extreme sets and the relative boundary actuallygoes deeper: Any face F of convex set C (that is not C itself) belongs torel ∂ C , so dim F < dim C . [194,18.1.3]2.8.2.2 Converse caveatIt is inconsequent to presume that each and every extreme point and directionis necessarily exposed, as might be erroneously inferred from the conventionalboundary definition (2.6.1.3.1); although it can correctly be inferred: eachand every extreme point and direction belongs to some exposed face.Arbitrary points residing on the relative boundary of a convex set are notnecessarily exposed or extreme points. Similarly, the direction of an arbitraryray, base 0, on the boundary of a convex cone is not necessarily an exposedor extreme direction. For the polyhedral cone illustrated in Figure 12, forexample, there are three two-dimensional exposed faces constituting theentire boundary, each composed of an infinity of rays. Yet there are onlythree exposed directions.Neither is an extreme direction on the boundary of a pointed convex conenecessarily an exposed direction. Lift the two-dimensional set in Figure 20,for example, into three dimensions such that no two points in the set arecollinear with the origin. Then its conic hull can have an extreme directionB on the boundary that is not an exposed direction, illustrated in Figure 28.