v2010.10.26 - Convex Optimization

2.13. DUAL CONE & GENERALIZED INEQUALITY 1852.13.6.0.1 Definition. Pointed polyhedral cone, vertex-description.Given pointed polyhedral cone K in a subspace of R n , denoting its i th extremedirection by Γ i ∈ R n arranged in a matrix X as in (280), then that cone maybe described: (86) (confer (188) (293))K = { [0 X ] aζ | a T 1 = 1, a ≽ 0, ζ ≥ 0 }{= Xaζ | a T 1 ≤ 1, a ≽ 0, ζ ≥ 0 }{ } (390)= Xb | b ≽ 0 ⊆ Rnthat is simply a conic hull (like (103)) of a finite number N of directions.Relative interior may always be expressedrel int K = {Xb | b ≻ 0} ⊂ R n (391)but identifying the cone’s relative boundary in this mannerrel∂K = {Xb | b ≽ 0, b ⊁ 0} (392)holds only when matrix X represents a bijection onto its range; in otherwords, some coefficients meeting lower bound zero (b∈∂R N +) do notnecessarily provide membership to relative boundary of cone K . △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 } (393)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 (297) to∀γ∈{Γ i } directly. 2.69 That dual cone so defined is unique, identical to (297),polyhedral whenever the number of generators N is finiteK ∗ = { y | X T y ≽ 0 } ⊆ R n (363)and is full-dimensional because K is assumed pointed.necessarily pointed unless K is full-dimensional (2.13.1.1).But K ∗ is not2.69 The extreme directions of K constitute a minimal set of generators. Formulae andconversions to vertex-description of the dual cone are in2.13.9 and2.13.11.