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Unformatted text preview: ametrizing the ellipsoid as E = {Bu + d  u 2 ≤ 1}, with B ∈ Sn and d ∈ Rn , the optimal
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ellipsoid can be found by solving the convex optimization problem
minimize − log det B
subject to B ai 2 + aT d ≤ bi ,
i i = 1, . . . , m with variables B ∈ Sn , d ∈ Rn . Derive the Lagrange dual of the equivalent problem
minimize − log det B
subject to yi 2 + aT d ≤ bi , i = 1, . . . , m
i
Bai = yi , i = 1, . . . , m
with variables B ∈ Sn , d ∈ Rn , yi ∈ Rn , i = 1, . . . , m.
7.16 Fitting a sphere to data. Consider the problem of ﬁtting a sphere {x ∈ Rn  x − xc
points u1 , . . . , um ∈ Rn , by minimizing the error function
m i=1 ui − x c 2
2 − r2 2 = r} to m 2 n over the variables xc ∈ R , r ∈ R.
(a) Explain how to solve this problem using convex or quasiconvex optimization. The simpler your
formulation, the better. (For example: a convex formulation is simpler than a quasiconvex
formulation; an LP is simpler than an SOCP, which is simpler than an SDP.) Be sure to explain
what your variables are, and how your fo...
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 Fall '13
 F.Borrelli
 The Aeneid

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