v2009.01.01 - Convex Optimization

2.13. DUAL CONE & GENERALIZED INEQUALITY 165
Whenever cone K is pointed closed and convex (not only polyhedral), then
dual cone K ∗ has a halfspace-description in terms of the extreme directions
Γ i of K :
K ∗ = { y | γ T y ≥ 0 for all γ ∈ {Γ i , i=1... N} ⊆ rel∂K } (347)
because when {Γ i } constitutes any set of generators for K , the discretization
result in2.13.4.1 allows relaxation of the requirement ∀x∈ K in (270) to
∀γ∈{Γ i } directly. 2.60 That dual cone so defined is unique, identical to (270),
polyhedral whenever the number of generators N is finite
K ∗ = { y | X T y ≽ 0 } ⊆ R n (327)
and has nonempty interior because K is assumed pointed (but K ∗ is not
necessarily pointed unless K has nonempty interior (2.13.1.1)).
2.13.6.1 Facet normal & extreme direction
We see from (327) that the conically independent generators of cone K
(namely, the extreme directions of pointed closed convex cone K constituting
the columns of X) each define an inward-normal to a hyperplane supporting
K ∗ (2.4.2.6.1) and exposing a dual facet when N is finite. Were K ∗
pointed and finitely generated, then by conjugation the dual statement
would also hold; id est, the extreme directions of pointed K ∗ each define
an inward-normal to a hyperplane supporting K and exposing a facet when
N is finite. Examine Figure 49 or Figure 54, for example.
We may conclude, the extreme directions of proper polyhedral K are
respectively orthogonal to the facets of K ∗ ; likewise, the extreme directions
of proper polyhedral K ∗ are respectively orthogonal to the facets of K .
2.13.7 Biorthogonal expansion by example
2.13.7.0.1 Example. Relationship to dual polyhedral cone.
Simplicial cone K illustrated in Figure 55 induces a partial order on R 2 . All
points greater than x with respect to K , for example, are contained in the
translated cone x + K . The extreme directions Γ 1 and Γ 2 of K do not make
2.60 The extreme directions of K constitute a minimal set of generators. Formulae and
conversions to vertex-descriptions of the dual cone are in2.13.9 and2.13.11.