bv_cvxbook_extra_exercises

Consider the geometric program minimize xt ay n

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Unformatted text preview: freely exchange products to produce, all together, the mix given by x. (The name ‘convolution’ presumably comes from the observation that if we replace the sum above with the product, and the infimum above with integration, then we obtain the normal convolution.) (a) Show that g is convex. ∗ ∗ (b) Show that g ∗ = f1 + · · · + fm . In other words, the conjugate of the infimal convolution is the sum of the conjugates. (c) Verify the identity in part (b) for the specific case of two strictly convex quadratic functions, fi (x) = (1/2)xT Pi x, with Pi ∈ Sn , i = 1, 2. ++ Hint: Depending on how you work out the conjugates, you might find the matrix identity (X + Y )−1 Y = X −1 (X −1 + Y −1 )−1 useful. 4.20 Derive the Lagrange dual of the optimization problem n φ(xi ) minimize i=1 subject to Ax = b with variable x ∈ Rn , where φ(u) = | u| c = −1 + , c − | u| c − | u| c is a positive parameter. The figure shows φ for c = 1. 36 dom φ = (−c, c). 5 4.5 4 3.5 φ(u) 3 2.5 2 1.5 1 0.5 0 −1 −0.5 0 u...
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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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