bv_cvxbook_extra_exercises

The transformed optimality conditions 6 are y 2 1

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

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.

Ask a homework question - tutors are online