v2009.01.01 - Convex Optimization

344 CHAPTER 4. SEMIDEFINITE PROGRAMMING
We find a good value of weight w ≈ 5 for randomized problems. Initial
value of direction matrix W is not critical. With dimension n = 26 and
number of equality constraints m = 10, convergence to a rank-ρ = 2 solution
to semidefinite feasibility problem (776) is achieved for nearly all realizations
in less than 10 iterations. (This finding is significant in so far as Barvinok’s
Proposition 2.9.3.0.1 predicts existence of matrices having rank 4 or less in
this intersection, but his upper bound can be no tighter than 3.) For small n ,
stall detection is required; one numerical implementation is disclosed on
Wıκımization.
This diagonalization decomposition technique is extensible to other
problem types, of course, including optimization problems having nontrivial
objectives. Because of a greatly expanded feasible set, for example,
find X ∈ S n
subject to A svec X ≽ b
X ≽ 0
rankX ≤ ρ
(785)
this relaxation of (776) [195,III] is more easily solved by convex
iteration rank-1. (The inequality in A remains in the constraints after
diagonalization.) Because of the nonconvex nature of a rank constrained
problem, more generally, there can be no proof of global convergence of convex
iteration from an arbitrary initialization; although, local convergence is
guaranteed by virtue of monotonically nonincreasing real objective sequences.
[222,1.2] [37,1.1]