v2009.01.01 - Convex Optimization

4.2. FRAMEWORK 255
sets are closed and convex, and the objective functions on a Euclidean vector
space are linear. Hence, convex optimization problems (584P) and (584D).
4.2.1.1 A ∩ S n + emptiness determination via Farkas’ lemma
4.2.1.1.1 Lemma. Semidefinite Farkas’ lemma.
Given an arbitrary set {A i ∈ S n , i=1... m} and a vector b = [b i ]∈ R m ,
define the affine subset
A = {X ∈ S n | 〈A i , X〉 = b i , i=1... m} (587)
Primal feasible set A ∩ S n + is nonempty if and only if y T b ≥ 0 holds for
each and every vector y = [y i ]∈ R m ∑
such that m y i A i ≽ 0.
Equivalently, primal feasible set A ∩ S n + is nonempty if and only
if y T b ≥ 0 holds for each and every norm-1 vector ‖y‖= 1 such that
m∑
y i A i ≽ 0.
⋄
i=1
Semidefinite Farkas’ lemma follows directly from a membership relation
(2.13.2.0.1) and the closed convex cones from linear matrix inequality
example 2.13.5.1.1; given convex cone K and its dual
where
i=1
K = {A svec X | X ≽ 0} (337)
m∑
K ∗ = {y | y j A j ≽ 0} (343)
⎡
A = ⎣
j=1
then we have membership relation
and equivalents
⎤
svec(A 1 ) T
. ⎦ ∈ R m×n(n+1)/2 (585)
svec(A m ) T
b ∈ K ⇔ 〈y,b〉 ≥ 0 ∀y ∈ K ∗ (288)
b ∈ K ⇔ ∃X ≽ 0 A svec X = b ⇔ A ∩ S n + ≠ ∅ (597)
b ∈ K ⇔ 〈y,b〉 ≥ 0 ∀y ∈ K ∗ ⇔ A ∩ S n + ≠ ∅ (598)