v2009.01.01 - Convex Optimization

180 CHAPTER 2. CONVEX GEOMETRY
Its dual is therefore pointed but of empty interior, having vertex-description
K ∗ M = {X ∗ b ∆ = [e 1 −e 2 e 2 −e 3 · · · e n−1 −e n ]b | b ≽ 0 } ⊂ R n (391)
where the columns of X ∗ comprise the extreme directions of K ∗ M . Because
K ∗ M is pointed and satisfies
rank(X ∗ ∈ R n×N ) = N ∆ = dim aff K ∗ ≤ n (392)
where N = n −1, and because K M is closed and convex, we may adapt Cone
Table 1 (p.174) as follows:
Cone Table 1* K ∗ K ∗∗ = K
vertex-description X ∗ X ∗†T , ±X ∗⊥
halfspace-description X ∗† , X ∗⊥T X ∗T
The vertex-description for K M is therefore
K M = {[X ∗†T X ∗⊥ −X ∗⊥ ]a | a ≽ 0} ⊂ R n (393)
where X ∗⊥ = 1 and
⎡
⎤
n − 1 −1 −1 · · · −1 −1 −1
n − 2 n − 2 −2
... · · · −2 −2
X ∗† = 1 . n − 3 n − 3 . .. −(n − 4) . −3
n
3 . n − 4 . ∈ R
.. n−1×n
−(n − 3) −(n − 3) .
⎢
⎥
⎣ 2 2 · · ·
... 2 −(n − 2) −(n − 2) ⎦
while
1 1 1 · · · 1 1 −(n − 1)
(394)
K ∗ M = {y ∈ R n | X ∗† y ≽ 0, X ∗⊥T y = 0} (395)
is the dual monotone cone halfspace-description.
2.13.9.4.4 Exercise. Inside the monotone cones.
Mathematically describe the respective interior of the monotone nonnegative
cone and monotone cone. In three dimensions, also describe the relative
interior of each face.