v2007.09.13 - Convex Optimization

172 CHAPTER 2. CONVEX GEOMETRYx 210.50K M−0.5−1K M−1.5K ∗ M−2−1 −0.5 0 0.5 1 1.5 2x 1Figure 51: Monotone cone K M and its dual K ∗ M (drawn truncated) in R2 .where the columns of X ∗ comprise the extreme directions of K ∗ M . BecauseK ∗ M is pointed and satisfiesrank(X ∗ ∈ R n×N ) = N ∆ = dim aff K ∗ ≤ n (379)where N = n−1, and because K M is closed and convex, we may adapt ConeTable 1 as follows:Cone Table 1* K ∗ K ∗∗ = Kvertex-description X ∗ X ∗†T , ±X ∗⊥halfspace-description X ∗† , X ∗⊥T X ∗TThe vertex-description for K M is thereforeK M = {[X ∗†T X ∗⊥ −X ∗⊥ ]a | a ≽ 0} ⊂ R n (380)where X ∗⊥ = 1 and⎡⎤n − 1 −1 −1 · · · −1 −1 −1n − 2 n − 2 −2... · · · −2 −2X ∗† = 1 . n − 3 n − 3 . .. −(n − 4) . −3n3 . n − 4 . ∈ R.. n−1×n−(n − 3) −(n − 3) .⎢⎥⎣ 2 2 · · ·... 2 −(n − 2) −(n − 2) ⎦1 1 1 · · · 1 1 −(n − 1)(381)