v2010.10.26 - Convex Optimization

368 CHAPTER 4. SEMIDEFINITE PROGRAMMINGTo transform max cut to its convex equivalent, first defineX = xx T ∈ S n (830)then max cut (822) becomesmaximize1X∈ S n 4〈X , δ(A1) − A〉subject to δ(X) = 1(X ≽ 0)rankX = 1(826)whose rank constraint can be regularized as inmaximize1X∈ S n 4subject to δ(X) = 1X ≽ 0〈X , δ(A1) − A〉 − w〈X , W 〉(827)where w ≈1000 is a nonnegative fixed weight, and W is a direction matrixdetermined fromn∑λ(X ⋆ ) ii=2= minimizeW ∈ S n 〈X ⋆ , W 〉subject to 0 ≼ W ≼ ItrW = n − 1(1700a)which has an optimal solution that is known in closed form. These twoproblems (827) and (1700a) are iterated until convergence as defined onpage 309.Because convex problem statement (827) is so elegant, it is numericallysolvable for large binary vectors within reasonable time. 4.41 To test ourconvex iterative method, we compare an optimal convex result to anactual solution of the max cut problem found by performing a brute forcecombinatorial search of (822) 4.42 for a tight upper bound. Search-time limitsbinary vector lengths to 24 bits (about five days CPU time). 98% accuracy,4.41 We solved for a length-250 binary vector in only a few minutes and convex iterationson a 2006 vintage laptop Core 2 CPU (Intel T7400@2.16GHz, 666MHz FSB).4.42 more computationally intensive than the proposed convex iteration by many ordersof magnitude. Solving max cut by searching over all binary vectors of length 100, forexample, would occupy a contemporary supercomputer for a million years.