v2007.09.13 - Convex Optimization

182 CHAPTER 2. CONVEX GEOMETRYhaving dim aff K = rankX = 3, then performing the most inefficientsimplicial decomposition in aff K we find⎡ ⎤ ⎡ ⎤1 1 01 1 0X 1 = ⎢ −1 0 1⎥⎣ 0 −1 0 ⎦ , X 2 = ⎢ −1 0 0⎥⎣ 0 −1 1 ⎦0 0 −10 0 −1X 3 =4X †T1 =⎢⎣⎡⎢⎣1 0 0−1 1 00 0 10 −1 −1⎤⎥⎦ , X 4 =⎡⎢⎣1 0 00 1 0−1 0 10 −1 −1⎤⎥⎦(414)The corresponding dual simplicial cones in aff K have generators respectivelycolumnar in⎡ ⎤ ⎡ ⎤2 1 1−2 1 1 ⎢4X †T3 =⎡⎢⎣Applying (412) we get2 −3 1−2 1 −33 2 −1−1 2 −1−1 −2 3−1 −2 −1⎥⎦ ,⎤⎥⎦ ,[ ]Γ ∗ 1 Γ ∗ 2 Γ ∗ 3 Γ ∗ 44X†T 2 =4X†T 4 =⎡= 1 ⎢4⎣⎢⎣⎡⎢⎣1 2 1−3 2 11 −2 11 −2 −33 −1 2−1 3 −2−1 −1 2−1 −1 −21 2 3 21 2 −1 −21 −2 −1 2−3 −2 −1 −2⎤⎥⎦⎤⎥⎦(415)⎥⎦ (416)whose rank is 3, and is the known result; 2.63 the conically independentgenerators for that pointed section of the dual cone K ∗ in aff K ; id est,K ∗ ∩ aff K .2.63 These calculations proceed so as to be consistent with [78,6]; as if the ambient vectorspace were the 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 statesthe dimension of the ordinary dual cone to be 3 ; it is, in fact, 4.