v2009.01.01 - Convex Optimization

2.13. DUAL CONE & GENERALIZED INEQUALITY 189
Applying algorithm (425) we get
[ ]
Γ ∗ 1 Γ ∗ 2 Γ ∗ 3 Γ ∗ 4
⎡
= 1 ⎢
4⎣
1 2 3 2
1 2 −1 −2
1 −2 −1 2
−3 −2 −1 −2
⎤
⎥
⎦ (429)
whose rank is 3, and is the known result; 2.73 a conically independent set
of generators for that pointed section of the dual cone K ∗ in aff K ; id est,
K ∗ ∩ aff K .
2.13.11.0.3 Example. Dual of proper polyhedral K in R 4 .
Given conically independent generators for a full-dimensional pointed closed
convex cone K
X = [ Γ 1 Γ 2 Γ 3 Γ 4 Γ 5 ] =
⎡
⎢
⎣
1 1 0 1 0
−1 0 1 0 1
0 −1 0 1 0
0 0 −1 −1 0
⎤
⎥
⎦ (430)
we count 5!/((5 −4)! 4!)=5 component simplices. Applying algorithm (425),
we find the six extreme directions of dual cone K ∗
⎡
⎤
1 0 0 1 1 1
[
]
Γ ∗ 1 Γ ∗ 2 Γ ∗ 3 Γ ∗ 4 Γ ∗ 5 Γ ∗ 6 = ⎢ 1 0 0 1 0 0
⎥
⎣ 1 0 −1 0 −1 1 ⎦ (431)
1 −1 −1 1 0 0
which means, (2.13.6.1) this proper polyhedral K = cone(X) has six
(three-dimensional) facets, whereas dual proper polyhedral cone K ∗ has only
five.
We can check this result (431) by reversing the process; we find
6!/((6 −4)! 4!) − 3=12 component simplices in the dual cone. 2.74 Applying
algorithm (425) to those simplices returns a conically independent set of
generators for K equivalent to (430).
2.73 These calculations proceed so as to be consistent with [97,6]; as if the ambient vector
space were proper subspace aff K whose dimension is 3. In that ambient space, K may
be regarded as a proper cone. Yet that author (from the citation) erroneously states
dimension of the ordinary dual cone to be 3 ; it is, in fact, 4.
2.74 Three combinations of four dual extreme directions are linearly dependent.