bv_cvxbook_extra_exercises

In other words the conjugate of the inmal convolution

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Unformatted text preview: fm (xm ) | x1 + · · · + xm = x}, with the natural domain (i.e., defined by g (x) < ∞). In one simple interpretation, fi (xi ) is the cost for the ith firm to produce a mix of products given by xi ; g (x) is then the optimal cost obtained if the firms can 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. 2.18 Conjugate of composition of convex and linear function. Suppose A ∈ Rm×n with rank A = m, and g is defined as g (x) = f (Ax), where f : Rm → R is convex. Show that g ∗ (y ) = f ∗ ((A† )T y ), dom(g ∗ ) = AT dom(f ∗ ), where A† = (AAT )−1 A is the pseudo-inverse of A. (This generaliz...
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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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