v2010.10.26 - Convex Optimization

296 CHAPTER 4. SEMIDEFINITE PROGRAMMINGeigenvalues,⎡λ(G ⋆ ) =⎢⎣6.999999777990990.000000226872410.000000022502960.00000000262974−0.00000000999738−0.00000000999875−0.00000001000000⎤⎥⎦(694)Negative eigenvalues are undoubtedly finite-precision effects. Because thelargest eigenvalue predominates by many orders of magnitude, we can expectto find a good approximation to a minimal cardinality Boolean solution bytruncating all smaller eigenvalues. We find, indeed, the desired result (683)⎛⎡x ⋆ = round⎜⎢⎝⎣0.000000001279470.000000005273690.000000001810010.999999974690440.000000014089500.00000000482903⎤⎞= e 4 (695)⎥⎟⎦⎠These numerical results are solver dependent; insofar, not all SDP solverswill return a rank-1 vertex solution.4.2.3.1.2 Example. Optimization over elliptope versus 1-norm polyhedronfor minimal cardinality Boolean Example 4.2.3.1.1.A minimal cardinality problem is typically formulated via, what is by now,the standard practice [120] [68,3.2,3.4] of column normalization applied toa 1-norm problem surrogate like (518). Suppose we define a diagonal matrix⎡Λ ⎢⎣⎤‖A(:,1)‖ 2 0‖A(:, 2)‖ 2 ⎥...0 ‖A(:, 6)‖ 2⎦ ∈ S6 (696)used to normalize the columns (assumed nonzero) of given noiseless datamatrix A . Then approximate the minimal cardinality Boolean problemminimize ‖x‖ 0xsubject to Ax = bx i ∈ {0, 1} ,i=1... n(680)