bv_cvxbook_extra_exercises

8 p t x 1 x 2 t x 3 t 2 x n t n 1 b the

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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.

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