bv_cvxbook_extra_exercises

98 093 y 2 101 101 1 00 0 0 1 0 p2

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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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