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Unformatted text preview: es the formula given on page 95
for the case when A is square and invertible.)
2.19 [5, p104] Suppose λ1 , . . . , λn are positive. Show that the function f : Rn → R, given by
n f (x) =
i=1 (1 − e−xi )λi , is concave on n dom f = x ∈ Rn
++ i=1 λi e− x i ≤ 1 . Hint. The Hessian is given by
∇2 f (x) = f (x)(yy T − diag(z ))
where yi = λi e−xi /(1 − e−xi ) and zi = yi /(1 − e−xi ).
2.20 Show that the following functions f : Rn → R are convex.
(a) The diﬀerence between the maximum and minimum value of a polynomial on a given interval,
as a function of its coeﬃcients:
f (x) = sup p(t) − inf p(t)
t∈[a,b] t∈[a,b] where a, b are real constants with a < b.
8 p( t ) = x 1 + x 2 t + x 3 t 2 + · · · + x n t n − 1 . (b) The ‘exponential barrier’ of a set of inequalities:
m f (x) = e−1/fi (x) , i=1 dom f = {x  fi (x) < 0, i = 1, . . . , m}. The functions fi are convex.
(c) The function
f (x) = inf α>0 g (y + αx) − g (y )
α if g is convex and y ∈ dom g . (It can be shown that this is the directional derivative of g at
y in the direction x....
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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 University of California, Berkeley.
 Fall '13
 F.Borrelli
 The Aeneid

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