v2009.01.01 - Convex Optimization

252 CHAPTER 4. SEMIDEFINITE PROGRAMMING
Now visualize intersection of the polyhedral cone with two (m = 2)
hyperplanes having linearly independent normals. The hyperplane
intersection A makes a line. Every intersecting line contains at least one
matrix having rank less than or equal to 2, providing an upper bound.
In other words, there exists a positive semidefinite matrix X belonging
to any line intersecting the polyhedral cone such that rankX ≤ 2.
In the case of three independent intersecting hyperplanes (m = 3), the
hyperplane intersection A makes a point that can reside anywhere in
the polyhedral cone. The upper bound on a point in S+ 3 is also the
greatest upper bound: rankX ≤ 3.
4.1.1.3.1 Example. Optimization on A ∩ S 3 + .
Consider minimization of the real linear function 〈C , X〉 on
a polyhedral feasible set;
P ∆ = A ∩ S 3 + (589)
f ⋆ 0
∆
= minimize 〈C , X〉
X
subject to X ∈ A ∩ S+
3
(590)
As illustrated for particular vector C and hyperplane A = ∂H in Figure 70,
this linear function is minimized on any X belonging to the face of P
containing extreme points {Γ 1 , Γ 2 } and all the rank-2 matrices in between;
id est, on any X belonging to the face of P
F(P) = {X | 〈C , X〉 = f ⋆ 0 } ∩ A ∩ S 3 + (591)
exposed by the hyperplane {X | 〈C , X〉=f ⋆ 0 }. In other words, the set of all
optimal points X ⋆ is a face of P
{X ⋆ } = F(P) = Γ 1 Γ 2 (592)
comprising rank-1 and rank-2 positive semidefinite matrices. Rank 1 is
the upper bound on existence in the feasible set P for this case m = 1
hyperplane constituting A . The rank-1 matrices Γ 1 and Γ 2 in face F(P)
are extreme points of that face and (by transitivity (2.6.1.2)) extreme
points of the intersection P as well. As predicted by analogy to Barvinok’s