v2009.01.01 - Convex Optimization

178 CHAPTER 2. CONVEX GEOMETRY
x 2
1
0.5
0
K M
−0.5
−1
K M
−1.5
K ∗ M
−2
−1 −0.5 0 0.5 1 1.5 2
x 1
Figure 57: Monotone cone K M and its dual K ∗ M (drawn truncated) in R2 .
Because X † x≽0 connotes membership of x to pointed K M+ , then by
(270) the dual cone we seek comprises all y for which (387) holds; thus its
halfspace-description
K ∗ M+ = {y ≽
K ∗ M+
0} = {y | ∑ k
i=1 y i ≥ 0, k = 1... n} = {y | X T y ≽ 0} ⊂ R n
(389)
The monotone nonnegative cone and its dual are simplicial, illustrated for
two Euclidean spaces in Figure 56.
From2.13.6.1, the extreme directions of proper K M+ are respectively
orthogonal to the facets of K ∗ M+ . Because K∗ M+ is simplicial, the
inward-normals to its facets constitute the linearly independent rows of X T
by (389). Hence the vertex-description for K M+ employs the columns of X
in agreement with Cone Table S because X † =X −1 . Likewise, the extreme
directions of proper K ∗ M+ are respectively orthogonal to the facets of K M+
whose inward-normals are contained in the rows of X † by (383). So the
vertex-description for K ∗ M+ employs the columns of X†T .
2.13.9.4.3 Example. Monotone cone.
(Figure 57, Figure 58) Of nonempty interior but not pointed, the monotone
cone is polyhedral and defined by the halfspace-description
K M ∆ = {x ∈ R n | x 1 ≥ x 2 ≥ · · · ≥ x n } = {x ∈ R n | X ∗T x ≽ 0} (390)