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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.
- Fall '13
- The Aeneid