v2009.01.01 - Convex Optimization

186 CHAPTER 2. CONVEX GEOMETRY
2.13.11 Proper nonsimplicial K , dual, X fat full-rank
Since conically dependent columns can always be removed from X to
construct K or to determine K ∗ (Wıκımization), then assume we are given
a set of N conically independent generators (2.10) of an arbitrary proper
polyhedral cone K in R n arranged columnar in X ∈ R n×N such that N > n
(fat) and rankX = n . Having found formula (374) to determine the dual of
a simplicial cone, the easiest way to find a vertex-description of proper dual
cone K ∗ is to first decompose K into simplicial parts K i so that K = ⋃ K i . 2.72
Each component simplicial cone in K corresponds to some subset of n
linearly independent columns from X . The key idea, here, is how the
extreme directions of the simplicial parts must remain extreme directions of
K . Finding the dual of K amounts to finding the dual of each simplicial part:
2.13.11.0.1 Theorem. Dual cone intersection. [286,2.7]
Suppose proper cone K ⊂ R n equals the union of M simplicial cones K i whose
extreme directions all coincide with those of K . Then proper dual cone K ∗
is the intersection of M dual simplicial cones K ∗ i ; id est,
M⋃
M⋂
K = K i ⇒ K ∗ = K ∗ i (418)
i=1
Proof. For X i ∈ R n×n , a complete matrix of linearly independent
extreme directions (p.130) arranged columnar, corresponding simplicial K i
(2.12.3.1.1) has vertex-description
i=1
K i = {X i c | c ≽ 0} (419)
⋄
Now suppose,
K =
M⋃
K i =
i=1
M⋃
{X i c | c ≽ 0} (420)
i=1
2.72 That proposition presupposes, of course, that we know how to perform simplicial
decomposition efficiently; also called “triangulation”. [262] [147,3.1] [148,3.1] Existence
of multiple simplicial parts means expansion of x∈ K like (365) can no longer be unique
because the number N of extreme directions in K exceeds dimension n of the space.