bv_cvxbook_extra_exercises

Assume that x is globally optimal for 5 we

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Unformatted text preview: d2 k 2 (4) − t = 0. The variables are x ∈ Rn , t ∈ R. Although this problem is not convex, it can be shown that strong duality holds. (It is a variation on the problem discussed on page 229 and in exercise 5.29 of Convex Optimization.) 27 Solve (4) for an example with m = 5, y1 = 1.8 2.5 , y2 = 2.0 1.7 , y3 = 1.5 1.5 , y4 = 1.5 2.0 , y5 = 2.5 1.5 , and d = (2.00, 1.24, 0.59, 1.31, 1.44). The figure shows some contour lines of the cost function f0 , with the positions yk indicated by circles. 3 2.5 x2 2 1.5 1 0.5 0.5 1 1.5 x1 2 2.5 3 To solve the problem, you can note that x⋆ is easily obtained from the KKT conditions for (4) if the optimal multiplier ν ⋆ for the equality constraint is known. You can use one of the following two methods to find ν ⋆ . • Derive the dual problem, express it as an SDP, and solve it using CVX. • Reduce the KKT conditions to a nonlinear equation in ν , and pick the correct solution (similarly as in exercise 5.29 of Convex Optimization ). 4.5 Projecti...
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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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