v2009.01.01 - Convex Optimization

4.2. FRAMEWORK 265
For the data given in (616), our semidefinite program solver (accurate in
solution to approximately 1E-8) 4.16 finds optimal solution to (626)
⎡
round(G ⋆ ) =
⎢
⎣
1 1 1 −1 1 1 −1
1 1 1 −1 1 1 −1
1 1 1 −1 1 1 −1
−1 −1 −1 1 −1 −1 1
1 1 1 −1 1 1 −1
1 1 1 −1 1 1 −1
−1 −1 −1 1 −1 −1 1
⎤
⎥
⎦
(627)
near a rank-1 vertex of the elliptope in S n+1 ; its sorted eigenvalues,
⎡
λ(G ⋆ ) =
⎢
⎣
6.99999977799099
0.00000022687241
0.00000002250296
0.00000000262974
−0.00000000999738
−0.00000000999875
−0.00000001000000
⎤
⎥
⎦
(628)
The negative eigenvalues are undoubtedly finite-precision effects. Because
the largest eigenvalue predominates by many orders of magnitude, we can
expect to find a good approximation to a minimum cardinality Boolean
solution by truncating all smaller eigenvalues. By so doing we find, indeed,
⎛⎡
x ⋆ = round
⎜⎢
⎝⎣
0.00000000127947
0.00000000527369
0.00000000181001
0.99999997469044
0.00000001408950
0.00000000482903
⎤⎞
= e 4 (629)
⎥⎟
⎦⎠
the desired result (617).
4.16 A typically ignored limitation of interior-point methods of solution is their relative
accuracy of only about 1E-8 on a machine using 64-bit (double precision) floating-point
arithmetic; id est, optimal solution x ⋆ cannot be more accurate than square root of
machine epsilon (ǫ=2.2204E-16). Duality gap is not a good measure of solution accuracy.