v2010.10.26 - Convex Optimization

178 CHAPTER 2. CONVEX GEOMETRY2.13.4.2.4 Example. Boundary membership to proper polyhedral cone.For a polyhedral cone, test (330) of boundary membership can be formulatedas a linear program. Say proper polyhedral cone K is specified completelyby generators that are arranged columnar inX = [ Γ 1 · · · Γ N ] ∈ R n×N (280)id est, K = {Xa | a ≽ 0}. Then membership relationc ∈ ∂K ⇔ ∃ y ≠ 0 〈y , c〉 = 0, y ∈ K ∗ , c ∈ K (330)may be expressedfinda , yy ≠ 0subject to c T y = 0X T y ≽ 0Xa = ca ≽ 0This linear feasibility problem has a solution iff c∈∂K .(371)2.13.4.3 smallest face of pointed closed convex coneGiven nonempty convex subset C of a convex set K , the smallest face of Kcontaining C is equivalent to intersection of all faces of K that contain C .[307, p.164] By (309), membership relation (330) means that each andevery point on boundary ∂K of proper cone K belongs to a hyperplanesupporting K whose normal y belongs to dual cone K ∗ . It follows that thesmallest face F , containing C ⊂ ∂K ⊂ R n on boundary of proper cone K ,is the intersection of all hyperplanes containing C whose normals are in K ∗ ;whereF(K ⊃ C) = {x ∈ K | x ⊥ K ∗ ∩ C ⊥ } (372)When C ∩ int K ≠ ∅ then F(K ⊃ C)= K .C ⊥ {y ∈ R n | 〈z, y〉=0 ∀z∈ C} (373)2.13.4.3.1 Example. Finding smallest face of cone.Suppose polyhedral cone K is completely specified by generators arrangedcolumnar inX = [ Γ 1 · · · Γ N ] ∈ R n×N (280)