v2009.01.01 - Convex Optimization

148 CHAPTER 2. CONVEX GEOMETRY
Conversely, if the closure of any convex cone K is pointed, then K ∗ has
nonempty interior;
K pointed ⇒ K ∗ nonempty interior (281)
Given that a cone K ⊂ R n is closed and convex, K is pointed if
and only if K ∗ − K ∗ = R n ; id est, iff K ∗
has nonempty interior.
[48,3.3, exer.20]
(vector sum) [266, thm.3.8] For convex cones K 1 and K 2
K 1 + K 2 = conv(K 1 ∪ K 2 ) (282)
(dual vector-sum) [266,16.4.2] [92,4.6] For convex cones K 1 and K 2
K ∗ 1 ∩ K ∗ 2 = (K 1 + K 2 ) ∗ = (K 1 ∪ K 2 ) ∗ (283)
(closure of vector sum of duals) 2.56 For closed convex cones K 1 and K 2
(K 1 ∩ K 2 ) ∗ = K ∗ 1 + K ∗ 2 = conv(K ∗ 1 ∪ K ∗ 2) (284)
where closure becomes superfluous under the condition K 1 ∩ int K 2 ≠ ∅
[48,3.3, exer.16,4.1, exer.7].
(Krein-Rutman) For closed convex cones K 1 ⊆ R m and K 2 ⊆ R n
and any linear map A : R n → R m , then provided int K 1 ∩ AK 2 ≠ ∅
[48,3.3.13, confer4.1, exer.9]
(A −1 K 1 ∩ K 2 ) ∗ = A T K ∗ 1 + K ∗ 2 (285)
where the dual of cone K 1 is with respect to its ambient space R m and
the dual of cone K 2 is with respect to R n , where A −1 K 1 denotes the
inverse image (2.1.9.0.1) of K 1 under mapping A , and where A T
denotes the adjoint operation.
2.56 These parallel analogous results for subspaces R 1 , R 2 ⊆ R n ; [92,4.6]
R ⊥⊥ = R for any subspace R.
(R 1 + R 2 ) ⊥ = R ⊥ 1 ∩ R ⊥ 2
(R 1 ∩ R 2 ) ⊥ = R ⊥ 1 + R⊥ 2