bv_cvxbook_extra_exercises

# Of course we have chosen an example in r2 so the

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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 ++ 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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