v2007.09.17 - Convex Optimization

284 CHAPTER 4. SEMIDEFINITE PROGRAMMINGwhose rank constraint can be regularized as inmaximize1X∈ S n 4subject to δ(X) = 1X ≽ 0〈X , δ(A1) − A〉 − w〈X , W 〉(684)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(1480a)whose optimal solution is known in closed form. These two problems (684)and (1480a) are iterated until convergence as defined on page 257.Because convex problem statement (684) is so elegant, it is numericallysolvable for large binary vectors within reasonable time. 4.29 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 (679) 4.30 for a tight upper bound. Search-time limitsbinary vector lengths to 24 bits (about five days cpu time). Accuracyobtained, 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. 4.31 A Matlab program is providedinF.7. That same accuracy is presumed maintained when binary vectorlength is further increased.4.29 We solved for a length-250 binary vector in only a few minutes and convex iterationson a Dell Precision model M90.4.30 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.4.31 Existence of a polynomial-time approximation to max cut with accuracy better than94.11% would refute proof of NP-hardness, which some researchers believe to be highlyunlikely. [131, thm.8.2]