v2010.10.26 - Convex Optimization

144 CHAPTER 2. CONVEX GEOMETRYhave no solution ζ ∈ R N + . That is ensured by conic independence of {y i ∈ R N }and by R(Y )∩ N(X) = 0 ; seen by factoring out X .2.10.1.1 linear maps of cones[21,7] If K is a convex cone in Euclidean space R and T is any linearmapping from R to Euclidean space M , then T(K) is a convex cone in Mand x ≼ y with respect to K implies T(x) ≼ T(y) with respect to T(K).If K is full-dimensional in R , then so is T(K) in M .If T is a linear bijection, then x ≼ y ⇔ T(x) ≼ T(y). If K is pointed,then so is T(K). And if K is closed, so is T(K). If F is a face of K , thenT(F) is a face of T(K).Linear bijection is only a sufficient condition for pointedness andclosedness; convex polyhedra (2.12) are invariant to any linear or inverselinear transformation [26,I.9] [307, p.44, thm.19.3].2.10.2 Pointed closed convex K & conic independenceThe following bullets can be derived from definitions (186) and (279) inconjunction with the extremes theorem (2.8.1.1.1):The set of all extreme directions from a pointed closed convex cone K ⊂ R nis not necessarily a linearly independent set, yet it must be a conicallyindependent set; (compare Figure 24 on page 71 with Figure 50a){extreme directions} ⇒ {c.i.}When a conically independent set of directions from pointed closed convexcone K is known to comprise generators, conversely, then all directions fromthat set must be extreme directions of the cone;{extreme directions} ⇔ {c.i. generators of pointed closed convex K}Barker & Carlson [21,1] call the extreme directions a minimal generatingset for a pointed closed convex cone. A minimal set of generators is thereforea conically independent set of generators, and vice versa, 2.52 for a pointedclosed convex cone.2.52 This converse does not hold for nonpointed closed convex cones as Table 2.10.0.0.1implies; e.g., ponder four conically independent generators for a plane (n=2, Figure 49).