v2007.09.13 - Convex Optimization

158 CHAPTER 2. CONVEX GEOMETRY2.13.6.0.1 Definition. Pointed polyhedral cone, vertex-description.(encore) (confer (252) (157)) Given pointed polyhedral cone K in a subspaceof R n , denoting its i th extreme direction by Γ i ∈ R n arranged in a matrix Xas in (240), then that cone may be described: (75) (confer (253))K = { [0 X ] aζ | a T 1 = 1, a ≽ 0, ζ ≥ 0 }{= Xaζ | a T 1 ≤ 1, a ≽ 0, ζ ≥ 0 }{ } (333)= Xb | b ≽ 0 ⊆ Rnthat is simply a conic hull (like (83)) of a finite number N of directions.△Whenever cone K is pointed closed and convex (not only polyhedral), thendual cone K ∗ has a halfspace-description in terms of the extreme directionsΓ i of K :K ∗ = { y | γ T y ≥ 0 for all γ ∈ {Γ i , i=1... N} ⊆ rel ∂K } (334)because when {Γ i } constitutes any set of generators for K , the discretizationresult in2.13.4.1 allows relaxation of the requirement ∀x∈ K in (258) to∀γ∈{Γ i } directly. 2.50 That dual cone so defined is unique, identical to (258),polyhedral whenever the number of generators N is finiteK ∗ = { y | X T y ≽ 0 } ⊆ R n (315)and has nonempty interior because K is assumed pointed (but K ∗ is notnecessarily pointed unless K has nonempty interior (2.13.1.1)).2.13.6.1 Facet normal & extreme directionWe see from (315) that the conically independent generators of cone K(namely, the extreme directions of pointed closed convex cone K constitutingthe columns of X) each define an inward-normal to a hyperplane supportingK ∗ (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 statementwould also hold; id est, the extreme directions of pointed K ∗ each definean inward-normal to a hyperplane supporting K and exposing a facet whenN is finite. Examine Figure 43 or Figure 48, for example.2.50 The extreme directions of K constitute a minimal set of generators.