bv_cvxbook_extra_exercises

2 minimize ii xb signz where z is the optimal value

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Unformatted text preview: ume that rank(A) = n, i.e., AT A is nonsingular. One possible application of this problem is as follows. A signal x ∈ {−1, 1}n is sent over a noisy ˆ channel, and received as b = Ax + v where v ∼ N (0, σ 2 I ) is Gaussian noise. The solution of (8) is ˆ the maximum likelihood estimate of the input signal x, based on the received signal b. ˆ (a) Derive the Lagrange dual of (8) and express it as an SDP. (b) Derive the dual of the SDP in part (a) and show that it is equivalent to minimize tr(AT AZ ) − 2bT Az + bT b subject to diag(Z ) = 1 Zz 0. zT 1 (9) Interpret this problem as a relaxation of (8). Show that if rank( Z zT z )=1 1 (10) at the optimum of (9), then the relaxation is exact, i.e., the optimal values of problems (8) and (9) are equal, and the optimal solution z of (9) is optimal for (8). This suggests a heuristic for rounding the solution of the SDP (9) to a feasible solution of (8), if (10) does not hold. We compute the eigenvalue decomposition Z zT n+1 z 1 vi ti λi = i=1 T vi ti , where vi ∈ Rn and ti ∈ R, and approximate the matrix by a rank...
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This note was uploaded on 09/10/2013 for the course C 231 taught by Professor F.borrelli during the Fall '13 term at University of California, Berkeley.

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