v2010.10.26 - Convex Optimization

164 CHAPTER 2. CONVEX GEOMETRY(vector sum) [307, thm.3.8] For convex cones K 1 and K 2K 1 + K 2 = conv(K 1 ∪ K 2 ) (312)is a convex cone.(dual vector-sum) [307,16.4.2] [109,4.6] For convex cones K 1 and K 2K ∗ 1 ∩ K ∗ 2 = (K 1 + K 2 ) ∗ = (K 1 ∪ K 2 ) ∗ (313)(closure of vector sum of duals) 2.62 For closed convex cones K 1 and K 2(K 1 ∩ K 2 ) ∗ = K ∗ 1 + K ∗ 2 = conv(K ∗ 1 ∪ K ∗ 2) (314)[330, p.96] where operation closure becomes superfluous under thesufficient condition K 1 ∩ int K 2 ≠ ∅ [55,3.3 exer.16,4.1 exer.7].(Krein-Rutman) Given 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 ≠ ∅[55,3.3.13, confer4.1 exer.9](A −1 K 1 ∩ K 2 ) ∗ = A T K ∗ 1 + K ∗ 2 (315)where dual of cone K 1 is with respect to its ambient space R m and dualof cone K 2 is with respect to R n , where A −1 K 1 denotes inverse image(2.1.9.0.1) of K 1 under mapping A , and where A T denotes adjointoperator. The particularly important case K 2 = R n is easy to show: forA T A = I(A T K 1 ) ∗ {y ∈ R n | x T y ≥ 0 ∀x∈A T K 1 }= {y ∈ R n | (A T z) T y ≥ 0 ∀z∈K 1 }= {A T w | z T w ≥ 0 ∀z∈K 1 }= A T K ∗ 1(316)2.62 These parallel analogous results for subspaces R 1 , R 2 ⊆ R n ; [109,4.6]R ⊥⊥ = R for any subspace R.(R 1 + R 2 ) ⊥ = R ⊥ 1 ∩ R ⊥ 2(R 1 ∩ R 2 ) ⊥ = R ⊥ 1 + R⊥ 2