bv_cvxbook_extra_exercises

Derive the dual problem and describe an ecient method

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Unformatted text preview: ct to equality constraints on x: n xk log(xk /yk ) minimize k=1 subject to Ax = b 1T x = 1 The optimization variable is x ∈ Rn . The domain of the objective function is Rn . The parameters ++ y ∈ Rn , A ∈ Rm×n , and b ∈ Rm are given. ++ Derive the Lagrange dual of this problem and simplify it to get maximize bT z − log n aT z k=1 yk e k (ak is the k th column of A). 4.4 Source localization from range measurements. [3] A signal emitted by a source at an unknown position x ∈ Rn (n = 2 or n = 3) is received by m sensors at known positions y1 , . . . , ym ∈ Rn . From the strength of the received signals, we can obtain noisy estimates dk of the distances x−yk 2 . We are interested in estimating the source position x based on the measured distances dk . In the following problem the error between the squares of the actual and observed distances is minimized: m minimize f0 (x) = k=1 x − yk 2 2 − d2 k 2 . Introducing a new variable t = xT x, we can express this as m minimize subject to k=1 xT x T t − 2yk x + yk 2 2...
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