v2007.09.13 - Convex Optimization

244 CHAPTER 4. SEMIDEFINITE PROGRAMMINGa desired solution resides on the elliptope relative boundary at a rank-1vertex. 4.14For the data given in (578), our semidefinite program solver (accurate toapproximately 1E-8) 4.15 finds optimal solution to (588)⎡round(G ⋆ ) =⎢⎣1 1 1 −1 1 1 −11 1 1 −1 1 1 −11 1 1 −1 1 1 −1−1 −1 −1 1 −1 −1 11 1 1 −1 1 1 −11 1 1 −1 1 1 −1−1 −1 −1 1 −1 −1 1near a rank-1 vertex of the elliptope in S n+1 ; its sorted eigenvalues,⎡⎤6.999999777990990.000000226872410.00000002250296λ(G ⋆ ) =0.00000000262974⎢ −0.00000000999738⎥⎣ −0.00000000999875 ⎦−0.00000001000000⎤⎥⎦(589)(590)The negative eigenvalues are undoubtedly finite-precision effects. Becausethe largest eigenvalue predominates by many orders of magnitude, we canexpect to find a good approximation to a minimum cardinality Booleansolution by truncating all smaller eigenvalues. By so doing we find, indeed,⎛⎡x ⋆ = round⎜⎢⎝⎣the desired result (579).0.000000001279470.000000005273690.000000001810010.999999974690440.000000014089500.00000000482903⎤⎞= e 4 (591)⎥⎟⎦⎠4.14 Confinement to the elliptope can be regarded as a kind of normalization akin tomatrix A column normalization suggested in [81] and explored in Example 4.2.3.0.3.4.15 A typically ignored limitation of interior-point methods of solution is their relativeaccuracy of only about 1E-8 on a machine using 64-bit (double precision) floating-pointarithmetic; id est, optimal solution cannot be more accurate than square root of machineepsilon (2.2204E-16).