v2007.09.13 - Convex Optimization

2.13. DUAL CONE & GENERALIZED INEQUALITY 181To find the extreme directions of the dual cone, first we observe that somefacets 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 ofK by design. For any particular polyhedral proper cone K , the extremedirections 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 amongthe inward-normals to facets of the component simplicial cones K i ; thosenormals are extreme directions of the dual simplicial cones K ∗ i . From thetheorem and Cone Table S (p.167),K ∗ =M⋂K ∗ i =i=1M⋂i=1{X †Ti c | c ≽ 0} (411)The set of extreme directions {Γ ∗ i } for proper dual cone K ∗ is thereforeconstituted by the conically independent generators, from the columnsof all the dual simplicial matrices {X †Ti } , that do not violate discretedefinition (315) of K ∗ ;{ }Γ ∗ 1 , Γ ∗ 2 ... Γ ∗ N{}= c.i. X †Ti (:,j), i=1... M , j =1... n | X † i (j,:)Γ l ≥ 0, l =1... N(412)where c.i. denotes selection of only the conically independent vectors fromthe argument set, argument (:,j) denotes the j th column while (j,:) denotesthe j th row, and {Γ l } constitutes the extreme directions of K . Figure 38(b)(p.123) shows a cone and its dual found via this formulation.2.13.11.0.2 Example. Dual of K nonsimplicial in subspace aff K .Given conically independent generators for pointed closed convex cone K inR 4 arranged columnar inX = [ Γ 1 Γ 2 Γ 3 Γ 4 ] =⎡⎢⎣1 1 0 0−1 0 1 00 −1 0 10 0 −1 −1⎤⎥⎦ (413)