v2010.10.26 - Convex Optimization

2.13. DUAL CONE & GENERALIZED INEQUALITY 2112.13.11.0.2 Example. Dual of K nonsimplicial in subspace aff K .Given conically independent generators for pointed closed convex cone K inR 4 arranged columnar inX = [ Γ 1 Γ 2 Γ 3 Γ 4 ] =⎡⎢⎣1 1 0 0−1 0 1 00 −1 0 10 0 −1 −1⎤⎥⎦ (478)having dim aff K = rankX = 3, (282) 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⎤⎥⎦(479)The corresponding dual simplicial cones in aff K have generators respectivelycolumnar in⎡ ⎤ ⎡ ⎤2 1 1−2 1 1 ⎢4X †T3 =⎡⎢⎣2 −3 1−2 1 −33 2 −1−1 2 −1−1 −2 3−1 −2 −1Applying algorithm (477) we get⎥⎦ ,⎤⎥⎦ ,[ ]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⎤⎥⎦⎤⎥⎦(480)⎥⎦ (481)