v2009.01.01 - Convex Optimization

316 CHAPTER 4. SEMIDEFINITE PROGRAMMING
Assign a rank-1 matrix of variables; symmetric variable matrix X and
solution vector x :
[ ] [ ] [ ]
x [ x
G =
T 1] X x ∆ xx =
1
x T =
T x
1 x T ∈ S N+1 (720)
1
Then design an equivalent semidefinite feasibility problem to find a Boolean
solution to Ax ≼ b :
find
X∈S N
x ∈ R N
subject to Ax ≼ b
G =
[ X x
x T 1
rankG = 1
(G ≽ 0)
δ(X) = 1
]
(721)
where x ⋆ i ∈ {−1, 1}, i=1... N . The two variables X and x are made
dependent via their assignment to rank-1 matrix G . By (1505), an optimal
rank-1 matrix G ⋆ must take the form (720).
As before, we regularize the rank constraint by introducing a direction
matrix Y into the objective:
minimize 〈G, Y 〉
X∈S N , x∈R N
subject to Ax ≼ b
[ X x
G =
x T 1
δ(X) = 1
]
≽ 0
(722)
Solution of this semidefinite program is iterated with calculation of the
direction matrix Y from semidefinite program (700). At convergence, in the
sense (671), convex problem (722) becomes equivalent to nonconvex Boolean
problem (719).
Direction matrix Y can be an orthogonal projector having closed-form
expression, by (1581a), although convex iteration is not a projection method.
Given randomized data A and b for a large problem, we find that
stalling becomes likely (convergence of the iteration to a positive fixed point