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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|
= −1 +
c − | u|
c − | u| c is a positive parameter. The ﬁgure shows φ for c = 1. 36 dom φ = (−c, c). 5
3.5 φ(u) 3
−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.
- Fall '13
- The Aeneid