v2010.10.26 - Convex Optimization

2.13. DUAL CONE & GENERALIZED INEQUALITY 199x 210.50K M−0.5−1K M−1.5K ∗ M−2−1 −0.5 0 0.5 1 1.5 2x 1Figure 65: Monotone cone K M and its dual K ∗ M (drawn truncated) in R2 .The monotone nonnegative cone and its dual are simplicial, illustrated fortwo Euclidean spaces in Figure 64.From2.13.6.1, the extreme directions of proper K M+ are respectivelyorthogonal to the facets of K ∗ M+ . Because K∗ M+ is simplicial, theinward-normals to its facets constitute the linearly independent rows of X Tby (435). Hence the vertex-description for K M+ employs the columns of Xin agreement with Cone Table S because X † =X −1 . Likewise, the extremedirections of proper K ∗ M+ are respectively orthogonal to the facets of K M+whose inward-normals are contained in the rows of X † by (429). So thevertex-description for K ∗ M+ employs the columns of X†T . 2.13.9.4.3 Example. Monotone cone.(Figure 65, Figure 66) Full-dimensional but not pointed, the monotone coneis polyhedral and defined by the halfspace-descriptionK M {x ∈ R n | x 1 ≥ x 2 ≥ · · · ≥ x n } = {x ∈ R n | X ∗T x ≽ 0} (436)Its dual is therefore pointed but not full-dimensional;K ∗ M = {X ∗ b [e 1 −e 2 e 2 −e 3 · · · e n−1 −e n ]b | b ≽ 0 } ⊂ R n (437)the dual cone vertex-description where the columns of X ∗ comprise itsextreme directions. Because dual monotone cone K ∗ M is pointed and satisfiesrank(X ∗ ∈ R n×N ) = N dim aff K ∗ ≤ n (438)