v2007.09.13 - Convex Optimization

142 CHAPTER 2. CONVEX GEOMETRY(dual vector-sum) [228,16.4.2] [73,4.6] For convex cones K 1 and K 2K ∗ 1 ∩ K ∗ 2 = (K 1 + K 2 ) ∗ = (K 1 ∪ K 2 ) ∗ (271)(closure of vector sum of duals) 2.47 For closed convex cones K 1 and K 2(K 1 ∩ K 2 ) ∗ = K ∗ 1 + K ∗ 2 = conv(K ∗ 1 ∪ K ∗ 2) (272)where closure becomes superfluous under the condition K 1 ∩ int K 2 ≠ ∅[41,3.3, exer.16,4.1, exer.7].(Krein-Rutman) For closed convex cones K 1 ⊆ R m and K 2 ⊆ R nand any linear map A : R n → R m , then provided int K 1 ∩ AK 2 ≠ ∅[41,3.3.13, confer4.1, exer.9](A −1 K 1 ∩ K 2 ) ∗ = A T K ∗ 1 + K ∗ 2 (273)where the dual of cone K 1 is with respect to its ambient space R m andthe dual of cone K 2 is with respect to R n , where A −1 K 1 denotes theinverse image (2.1.9.0.1) of K 1 under mapping A , and where A Tdenotes the adjoint operation.K is proper if and only if K ∗ is proper.K is polyhedral if and only if K ∗ is polyhedral. [245,2.8]K is simplicial if and only if K ∗ is simplicial. (2.13.9.2) A simplicialcone and its dual are polyhedral proper cones (Figure 50, Figure 41),but not the converse.K ⊞ −K ∗ = R n ⇔ K is closed and convex. (1787) (p.674)Any direction in a proper cone K is normal to a hyperplane separatingK from −K ∗ .2.47 These parallel analogous results for subspaces R 1 , R 2 ⊆ R n ; [73,4.6]R ⊥⊥ = R for any subspace R.(R 1 + R 2 ) ⊥ = R ⊥ 1 ∩ R ⊥ 2(R 1 ∩ R 2 ) ⊥ = R ⊥ 1 + R⊥ 2