v2009.01.01 - Convex Optimization

322 CHAPTER 4. SEMIDEFINITE PROGRAMMING
whose rank constraint can be regularized as in
maximize
1
X∈ S n 4
subject to δ(X) = 1
X ≽ 0
〈X , δ(A1) − A〉 − w〈X , W 〉
(738)
where w ≈1000 is a nonnegative fixed weight, and W is a direction matrix
determined from
n∑
λ(X ⋆ ) i
i=2
= minimize
W ∈ S n 〈X ⋆ , W 〉
subject to 0 ≼ W ≼ I
trW = n − 1
(1581a)
which has an optimal solution that is known in closed form. These two
problems (738) and (1581a) are iterated until convergence as defined on
page 278.
Because convex problem statement (738) is so elegant, it is numerically
solvable for large binary vectors within reasonable time. 4.35 To test our
convex iterative method, we compare an optimal convex result to an
actual solution of the max cut problem found by performing a brute force
combinatorial search of (733) 4.36 for a tight upper bound. Search-time limits
binary vector lengths to 24 bits (about five days CPU time). Accuracy
obtained, 98%, is independent of binary vector length (12, 13, 20, 24)
when averaged over more than 231 problem instances including planar,
randomized, and toroidal graphs. 4.37 When failure occurred, large and
small errors were manifest. That same 98% average accuracy is presumed
maintained when binary vector length is further increased. A Matlab
program is provided on Wıκımization.
4.35 We solved for a length-250 binary vector in only a few minutes and convex iterations
on a Dell Precision computer, model M90.
4.36 more computationally intensive than the proposed convex iteration by many orders
of magnitude. Solving max cut by searching over all binary vectors of length 100, for
example, would occupy a contemporary supercomputer for a million years.
4.37 Existence of a polynomial-time approximation to max cut with accuracy better than
94.11% would refute proof of NP-hardness, which Håstad believes to be highly unlikely.
[155, thm.8.2] [156]