v2009.01.01 - Convex Optimization

254 CHAPTER 4. SEMIDEFINITE PROGRAMMING
with an equal number of twos and zeros along the main diagonal. Indeed,
optimal solution (595) is a terminal solution along the central path taken by
the interior-point method as implemented in [342,2.5.3]; it is also a solution
of highest rank among all optimal solutions to (594). Clearly, rank of this
primal optimal solution exceeds by far a rank-1 solution predicted by upper
bound (245).
4.1.1.5 Later developments
This rational example (594) indicates the need for a more generally applicable
and simple algorithm to identify an optimal solution X ⋆ satisfying Barvinok’s
Proposition 2.9.3.0.1. We will review such an algorithm in4.3, but first we
provide more background.
4.2 Framework
4.2.1 Feasible sets
Denote by C and C ∗ the convex sets of primal and dual points respectively
satisfying the primal and dual constraints in (584), each assumed nonempty;
⎧ ⎡ ⎤ ⎫
⎨
〈A 1 , X〉 ⎬
C =
⎩ X ∈ Sn + | ⎣ . ⎦= b
⎭ = A ∩ Sn +
〈A m , X〉
(596)
{
}
m∑
C ∗ = S ∈ S n + , y = [y i ]∈ R m | y i A i + S = C
These are the primal feasible set and dual feasible set. Geometrically,
primal feasible A ∩ S n + represents an intersection of the positive semidefinite
cone S n + with an affine subset A of the subspace of symmetric matrices S n
in isometrically isomorphic R n(n+1)/2 . The affine subset has dimension
n(n+1)/2 −m when the A i are linearly independent. Dual feasible set C ∗
is the Cartesian product of the positive semidefinite cone with its inverse
image (2.1.9.0.1) under affine transformation C − ∑ y i A i . 4.6 Both feasible
4.6 The inequality C − ∑ y i A i ≽0 follows directly from (584D) (2.9.0.1.1) and is known
as a linear matrix inequality. (2.13.5.1.1) Because ∑ y i A i ≼C , matrix S is known as a
slack variable (a term borrowed from linear programming [82]) since its inclusion raises
this inequality to equality.
i=1