v2009.01.01 - Convex Optimization

2.13. DUAL CONE & GENERALIZED INEQUALITY 187
The union of all K i can be equivalently expressed
⎧
⎡ ⎤
⎪⎨
K =
[X 1 X 2 · · · X M ] ⎢
⎣
⎪⎩
a
ḅ
.
c
⎥
⎦ | a , b ... c ≽ 0 ⎫⎪ ⎬
⎪ ⎭
(421)
Because extreme directions of the simplices K i are extreme directions of K
by assumption, then by the extremes theorem (2.8.1.1.1),
K = { [ X 1 X 2 · · · X M ]d | d ≽ 0 } (422)
Defining X ∆ = [X 1 X 2 · · · X M ] (with any redundant [sic] columns optionally
removed from X), then K ∗ can be expressed, (327) (Cone Table S, p.174)
K ∗ = {y | X T y ≽ 0} =
M⋂
{y | Xi T y ≽ 0} =
i=1
M⋂
K ∗ i (423)
i=1
To find the extreme directions of the dual cone, first we observe that some
facets of each simplicial part K i are common to facets of K by assumption,
and the union of all those common facets comprises the set of all facets of
K by design. For any particular proper polyhedral cone K , the extreme
directions of dual cone K ∗ are respectively orthogonal to the facets of K .
(2.13.6.1) Then the extreme directions of the dual cone can be found
among inward-normals to facets of the component simplicial cones K i ; those
normals are extreme directions of the dual simplicial cones K ∗ i . From the
theorem and Cone Table S (p.174),
K ∗ =
M⋂
K ∗ i =
i=1
M⋂
i=1
{X †T
i c | c ≽ 0} (424)
The set of extreme directions {Γ ∗ i } for proper dual cone K ∗ is therefore
constituted by those conically independent generators, from the columns
of all the dual simplicial matrices {X †T
i } , that do not violate discrete
definition (327) of K ∗ ;
{ } {
}
Γ ∗ 1 , Γ ∗ 2 ... Γ ∗ N = c.i. X †T
i (:,j), i=1... M , j =1... n | X † i (j,:)Γ l ≥ 0, l =1... N
(425)