bv_cvxbook_extra_exercises

# The transformed optimality conditions 6 are y 2 1

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Unformatted text preview: on on the ℓ1 ball. Consider the problem of projecting a point a ∈ Rn on the unit ball in ℓ1 -norm: minimize (1/2) x − a 2 2 subject to x 1 ≤ 1. Derive the dual problem and describe an eﬃcient method for solving it. Explain how you can obtain the optimal x from the solution of the dual problem. 4.6 A nonconvex problem with strong duality. On page 229 of Convex Optimization, we consider the problem minimize f (x) = xT Ax + 2bT x (5) subject to xT x ≤ 1 28 with variable x ∈ Rn , and data A ∈ Sn , b ∈ Rn . We do not assume that A is positive semideﬁnite, and therefore the problem is not necessarily convex. In this exercise we show that x is (globally) optimal if and only if there exists a λ such that x 2 ≤ 1, λ ≥ 0, A + λI (A + λI )x = −b, 0, λ(1 − x 2 ) = 0. 2 (6) From this we will develop an eﬃcient method for ﬁnding the global solution. The conditions (6) are the KKT conditions for (5) with the inequality A + λI 0 added. (a) Show that if x and λ satisfy...
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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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