v2010.10.26 - Convex Optimization

2.8. CONE BOUNDARY 1112.8.2 Exposed direction2.8.2.0.1 Definition. Exposed point & direction of pointed convex cone.[307,18] (confer2.6.1.0.1)When a convex cone has a vertex, an exposed point, it resides at theorigin; there can be only one.In the closure of a pointed convex cone, an exposed direction is thedirection of a one-dimensional exposed face that is a ray emanatingfrom the origin.{exposed directions} ⊆ {extreme directions}△For a proper cone in vector space R n with n ≥ 2, we can say more:{exposed directions} = {extreme directions} (189)It follows from Lemma 2.8.0.0.1 for any pointed closed convex cone, thereis one-to-one correspondence of one-dimensional exposed faces with exposeddirections; id est, there is no one-dimensional exposed face that is not a raybase 0.The pointed closed convex cone EDM 2 , for example, is a ray inisomorphic subspace R whose relative boundary (2.6.1.4.1) is the origin.The conventionally exposed directions of EDM 2 constitute the empty set∅ ⊂ {extreme direction}. This cone has one extreme direction belonging toits relative interior; an idiosyncrasy of dimension 1.2.8.2.1 Connection between boundary and extremes2.8.2.1.1 Theorem. Exposed. [307,18.7] (confer2.8.1.1.1)Any closed convex set C containing no lines (and whose dimension is atleast 2) can be expressed as closure of the convex hull of its exposed pointsand exposed rays.⋄From Theorem 2.8.1.1.1,rel ∂ C = C \ rel int C (24)= conv{exposed points and exposed rays} \ rel int C= conv{extreme points and extreme rays} \ rel int C⎫⎪⎬⎪⎭(190)