v2009.01.01 - Convex Optimization

256 CHAPTER 4. SEMIDEFINITE PROGRAMMING
Semidefinite Farkas’ lemma provides the conditions required for a set
of hyperplanes to have a nonempty intersection A ∩ S n + with the positive
semidefinite cone. While the lemma as stated is correct, Ye points out
[342,1.3.8] that a positive definite version of this lemma is required for
semidefinite programming because any feasible point in the relative interior
A ∩ int S n + is required by Slater’s condition 4.7 to achieve 0 duality gap
(primal−dual objective difference4.2.3, Figure 51). In our circumstance,
assuming a nonempty intersection, a positive definite lemma is required
to insure a point of intersection closest to the origin is not at infinity;
e.g., Figure 39. Then given A∈ R m×n(n+1)/2 having rank m , we wish to
detect existence of a nonempty relative interior of the primal feasible set; 4.8
b ∈ int K ⇔ 〈y, b〉 > 0 ∀y ∈ K ∗ , y ≠ 0 ⇔ A∩int S n + ≠ ∅ (599)
A positive definite Farkas’ lemma can easily be constructed from this
membership relation (294) and these proper convex cones K (337) and
K ∗ (343):
4.2.1.1.2 Lemma. Positive definite Farkas’ lemma.
Given a linearly independent set {A i ∈ S n , i=1... m}
b = [b i ]∈ R m , define the affine subset
and a vector
A = {X ∈ S n | 〈A i , X〉 = b i , i=1... m} (587)
Primal feasible set relative interior A ∩ int S n + is nonempty if and only if
∑
y T b > 0 holds for each and every vector y = [y i ]≠ 0 such that m y i A i ≽ 0.
Equivalently, primal feasible set relative interior A ∩ int 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
4.7 Slater’s sufficient condition is satisfied whenever any primal strictly feasible point
exists; id est, any point feasible with the affine equality (or affine inequality) constraint
functions and relatively interior to convex cone K . [286,6.6] [36, p.325] If cone K were
polyhedral, then Slater’s condition is satisfied when any feasible point exists relatively
interior to K or on its relative boundary. [53,5.2.3]
4.8 Detection of A ∩ int S n + by examining K interior is a trick need not be lost.
i=1