bv_cvxbook_extra_exercises

# Therefore xk converges to 0 0 however this is not the

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Unformatted text preview: rmulation minimizes the error function above. (b) Use your method to solve the problem instance with data given in the ﬁle sphere_fit_data.m, with n = 2. Plot the ﬁtted circle and the data points. 7.17 The polar of a set C ⊆ Rn is deﬁned as C ◦ = {x | uT x ≤ 1 ∀u ∈ C }. (a) Show that C ◦ is convex, regardless of the properties of C . (b) Let C1 and C2 be two nonempty polyhedra deﬁned by sets of linear inequalities: C1 = {u ∈ Rn | A1 u b1 }, C2 = {v ∈ Rn | A2 v b2 } with A1 ∈ Rm1 ×n , A2 ∈ Rm2 ×n , b1 ∈ Rm1 , b2 ∈ Rm2 . Formulate the problem of ﬁnding the ◦ ◦ Euclidean distance between C1 and C2 , minimize x1 − x2 ◦ subject to x1 ∈ C1 ◦ x 2 ∈ C2 , 2 2 as a QP. Your formulation should be eﬃcient, i.e., the dimensions of the QP (number of variables and constraints) should be linear in m1 , m2 , n. (In particular, formulations that require enumerating the extreme points of C1 and C2 are to be avoided.) 71 7.18 Polyhedral cone questions. You are given matrices A ∈ Rn×k and B ∈ Rn×p . Explai...
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## This note was uploaded on 09/10/2013 for the course C 231 taught by Professor F.borrelli during the Fall '13 term at Berkeley.

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