v2010.10.26 - Convex Optimization

212 CHAPTER 2. CONVEX GEOMETRYwhose rank is 3, and is the known result; 2.85 a conically independent setof generators for that pointed section of the dual cone K ∗ in aff K ; id est,K ∗ ∩ aff K .2.13.11.0.3 Example. Dual of proper polyhedral K in R 4 .Given conically independent generators for a full-dimensional pointed closedconvex cone KX = [ Γ 1 Γ 2 Γ 3 Γ 4 Γ 5 ] = ⎢⎣⎡1 1 0 1 0−1 0 1 0 10 −1 0 1 00 0 −1 −1 0⎤⎥⎦ (482)we count 5!/((5 −4)! 4!)=5 component simplices. 2.86 Applying algorithm(477), we find the six extreme directions of dual cone K ∗ (with Γ 2 = Γ ∗ 5)X ∗ =[]Γ ∗ 1 Γ ∗ 2 Γ ∗ 3 Γ ∗ 4 Γ ∗ 5 Γ ∗ 6=⎡⎢⎣1 0 0 1 1 11 0 0 1 0 01 0 −1 0 −1 11 −1 −1 1 0 0⎤⎥⎦ (483)which means, (2.13.6.1) this proper polyhedral K = cone(X) has six(three-dimensional) facets generated G by its {extreme directions}:⎧ ⎧F 1F 2 ⎪⎨⎫⎪ ⎬ ⎪⎨FG 3=F 4F ⎪⎩ 5 ⎪ ⎭ ⎪⎩F 6Γ 1 Γ 2 Γ 3Γ 1 Γ 2 Γ 5⎫⎪ ⎬Γ 1 Γ 4 Γ 5(484)Γ 1 Γ 3 Γ 4Γ 3 Γ 4 Γ 5 ⎪ ⎭Γ 2 Γ 3 Γ 52.85 These calculations proceed so as to be consistent with [114,6]; as if the ambientvector space were proper subspace aff K whose dimension is 3. In that ambient space, Kmay be regarded as a proper cone. Yet that author (from the citation) erroneously statesdimension of the ordinary dual cone to be 3 ; it is, in fact, 4.2.86 There are no linearly dependent combinations of three or four extreme directions inthe primal cone.