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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
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y ∈ Rn , A ∈ Rm×n , and b ∈ Rm are given.
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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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 Fall '13
 F.Borrelli
 The Aeneid

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