v2009.01.01 - Convex Optimization

4.4. RANK-CONSTRAINED SEMIDEFINITE PROGRAM 277
Alternatively, we say the i th inequality constraint is active when it is
met with equality; id est, when for particular i in (667), αi T svec X ⋆ = β i .
An optimal high-rank solution X ⋆ is, of course, feasible satisfying all the
constraints. But for the purpose of rank reduction, inactive inequality
constraints are ignored while active inequality constraints are interpreted as
equality constraints. In other words, we take the union of active inequality
constraints (as equalities) with equality constraints A svec X = b to form
a composite affine subset  substituting for (587). Then we proceed with
rank reduction of X ⋆ as though the semidefinite program were in prototypical
form (584P).
4.4 Rank-constrained semidefinite program
A technique for finding low-rank optimal solutions to semidefinite programs
of a more general form was discovered in October 2005:
4.4.1 rank constraint by convex iteration
Consider a semidefinite feasibility problem of the form
find
G∈S N
G
subject to G ∈ C
G ≽ 0
rankG ≤ n
(669)
where C is a convex set presumed to contain positive semidefinite matrices
of rank n or less; id est, C intersects the positive semidefinite cone boundary.
We propose that this rank-constrained feasibility problem is equivalently
expressed as the problem sequence (670) (1581a) having convex constraints:
minimize
G∈S N 〈G , W 〉
subject to G ∈ C
G ≽ 0
(670)
where direction matrix 4.21 W is an optimal solution to semidefinite program,
for 0≤n≤N −1
4.21 Search direction W is a hyperplane-normal pointing opposite to direction of movement
describing minimization of a real linear function 〈G, W 〉 (p.69).