bv_cvxbook_extra_exercises

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Unformatted text preview: nctions fi are convex and diﬀerentiable. For u > p⋆ , deﬁne xac (u) as the analytic center of the inequalities f0 (x) ≤ u, fi (x) ≤ 0, i = 1, . . . , m, i.e., m xac (u) = argmin − log(u − f0 (x)) − i=1 log(−fi (x)) . Show that λ ∈ Rm , deﬁned by λi = u − f0 (xac (u)) , −fi (xac (u)) i = 1, . . . , m is dual feasible for the problem above. Express the corresponding dual objective value in terms of u, xac (u) and the problem parameters. 9.2 Eﬃcient solution of Newton equations. Explain how you would solve the Newton equations in the barrier method applied to the quadratic program minimize (1/2)xT x + cT x subject to Ax b where A ∈ Rm×n is dense. Distinguish two cases, m ≫ n and n ≫ m, and give the most eﬃcient method in each case. 9.3 Eﬃcient solution of Newton equations. Describe an eﬃcient method for solving the Newton equation in the barrier method for the quadratic program minimize (1/2)(x − a)T P −1 (x − a) subject to 0 x 1, with variable x ∈ Rn . The matrix P ∈ Sn and the vector a ∈ Rn are given. As...
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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 University of California, Berkeley.

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