v2009.01.01 - Convex Optimization

188 CHAPTER 2. CONVEX GEOMETRY
where c.i. denotes selection of only the conically independent vectors from
the argument set, argument (:,j) denotes the j th column while (j,:) denotes
the j th row, and {Γ l } constitutes the extreme directions of K . Figure 44(b)
(p.128) shows a cone and its dual found via this algorithm.
2.13.11.0.2 Example. Dual of K nonsimplicial in subspace aff K .
Given conically independent generators for pointed closed convex cone K in
R 4 arranged columnar in
X = [ Γ 1 Γ 2 Γ 3 Γ 4 ] =
⎡
⎢
⎣
1 1 0 0
−1 0 1 0
0 −1 0 1
0 0 −1 −1
⎤
⎥
⎦ (426)
having dim aff K = rankX = 3, (254) then performing the most inefficient
simplicial decomposition in aff K we find
X 1 =
X 3 =
4X †T
1 =
⎢
⎣
⎡
⎢
⎣
⎡
⎢
⎣
1 1 0
−1 0 1
0 −1 0
0 0 −1
1 0 0
−1 1 0
0 0 1
0 −1 −1
⎤
⎥
⎦ , X 2 =
⎤
⎥
⎦ , X 4 =
⎡
⎢
⎣
⎡
⎢
⎣
1 1 0
−1 0 0
0 −1 1
0 0 −1
1 0 0
0 1 0
−1 0 1
0 −1 −1
⎤
⎥
⎦
⎤
⎥
⎦
(427)
The corresponding dual simplicial cones in aff K have generators respectively
columnar in
⎡ ⎤ ⎡ ⎤
2 1 1
−2 1 1 ⎢
4X †T
3 =
⎡
⎢
⎣
2 −3 1
−2 1 −3
3 2 −1
−1 2 −1
−1 −2 3
−1 −2 −1
⎥
⎦ ,
⎤
⎥
⎦ ,
4X†T 2 =
4X†T 4 =
⎢
⎣
⎡
⎢
⎣
1 2 1
−3 2 1
1 −2 1
1 −2 −3
3 −1 2
−1 3 −2
−1 −1 2
−1 −1 −2
⎥
⎦
⎤
⎥
⎦
(428)