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Unformatted text preview: ickest slab separating two sets. We are given two sets in Rn : a polyhedron
C1 = {x  Cx d}, deﬁned by a matrix C ∈ Rm×n and a vector d ∈ Rm , and an ellipsoid
C2 = {P u + q  u 2 ≤ 1}, deﬁned by a matrix P ∈ Rn×n and a vector q ∈ Rn . We assume that the sets are nonempty and
that they do not intersect. We are interested in the optimization problem
maximize inf x∈C1 aT x − supx∈C2 aT x
subject to a 2 = 1.
with variable a ∈ Rn . Explain how you would solve this problem. You can answer the question by reducing the problem
to a standard problem class (LP, QP, SOCP, SDP, . . . ), or by describing an algorithm to solve it.
Remark. The geometrical interpretation is as follows. If we choose
1
b = ( inf aT x + sup aT x),
2 x ∈C 1
x ∈C 2
then the hyperplane H = {x  aT x = b} is the maximum margin separating hyperplane separating
C1 and C2 . Alternatively, a gives us the thickest slab that separates the two sets. 7.8 Bounding object position from multiple camera views. A...
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 Fall '13
 F.Borrelli
 The Aeneid

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