v2009.01.01 - Convex Optimization

4.2. FRAMEWORK 267
minimizes an affine function on an intersection of the elliptope with
hyperplanes. Although the same Boolean solution is obtained from
this approximation (631) as compared with semidefinite program (626),
when given that particular data from Example 4.2.3.1.1, Singer confides a
counter-example: Instead, given data
[ ] [ ]
1 0 √2 1
1
A =
1
, b =
(633)
0 1 √2 1
then solving approximation (631) yields
⎛⎡
1 − √ 1
2
y ⋆ ⎜⎢
= round⎝⎣
1 − √ 1
2
1
⎤⎞
⎥⎟
⎦⎠ =
⎡
⎢
⎣
0
0
1
⎤
⎥
⎦ (634)
(infeasible, with or without rounding, with respect to original problem (614))
whereas solving semidefinite program (626) produces
⎡
⎤
1 1 −1 1
round(G ⋆ ) = ⎢ 1 1 −1 1
⎥
⎣ −1 −1 1 −1 ⎦ (635)
1 1 −1 1
with sorted eigenvalues
λ(G ⋆ ) =
⎡
⎢
⎣
3.99999965057264
0.00000035942736
−0.00000000000000
−0.00000001000000
⎤
⎥
⎦ (636)
Truncating all but the largest eigenvalue, from (622) we obtain (confer y ⋆ )
⎛⎡
⎤⎞
⎡ ⎤
0.99999999625299 1
x ⋆ = round⎝⎣
0.99999999625299 ⎦⎠ = ⎣ 1 ⎦ (637)
0.00000001434518 0
the desired minimum cardinality Boolean result.
4.2.3.1.3 Exercise. Minimum cardinality Boolean art.
Assess general performance of standard-practice approximation (631) as
compared with the proposed semidefinite program (626).