v2007.09.17 - Convex Optimization

246 CHAPTER 4. SEMIDEFINITE PROGRAMMINGminimizes an affine function on the elliptope intersected byhyperplanes. Although the same Boolean solution is obtained fromthis approximation (593) as compared with semidefinite program (588)when given that particular data from Example 4.2.3.0.2, Singer confides acounter-example: Instead, given data[ ] [ ]1 0 √2 11A =1, b =(595)0 1 √2 1then solving approximation (593) yields⎛⎡1 − √ 12y ⋆ ⎜⎢= round⎝⎣1 − √ 121⎤⎞⎥⎟⎦⎠ =⎡⎢⎣001⎤⎥⎦ (596)(infeasible, with or without rounding, with respect to original problem (576))whereas solving semidefinite program (588) produces⎡⎤1 1 −1 1round(G ⋆ ) = ⎢ 1 1 −1 1⎥⎣ −1 −1 1 −1 ⎦ (597)1 1 −1 1with sorted eigenvaluesλ(G ⋆ ) =⎡⎢⎣3.999999650572640.00000035942736−0.00000000000000−0.00000001000000⎤⎥⎦ (598)Truncating all but the largest eigenvalue, we obtain (confer y ⋆ ) (584)⎛⎡⎤⎞⎡ ⎤0.99999999625299 1x ⋆ = round⎝⎣0.99999999625299 ⎦⎠ = ⎣ 1 ⎦ (599)0.00000001434518 0the desired minimum cardinality Boolean result.We leave pending a general performance assessment of standard-practiceapproximation (593) as compared with our proposed semidefiniteprogram (588).