bv_cvxbook_extra_exercises

hint to nd the maximum of y t x over x c write the

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Unformatted text preview: y. Suppose the random variable X on Rn has log-concave density, and let Y = g (X ), where g : Rn → R. For each of the following statements, either give a counterexample, or show that the statement is true. (a) If g is aﬃne and not constant, then Y has log-concave density. (b) If g is convex, then prob(Y ≤ a) is a log-concave function of a. (c) If g is concave, then E ((Y − a)+ ) is a convex and log-concave function of a. (This quantity is called the tail expectation of Y ; you can assume it exists. We deﬁne (s)+ as (s)+ = max{s, 0}.) 2.28 Majorization. Deﬁne C as the set of all permutations of a given n-vector a, i.e., the set of vectors (aπ1 , aπ2 , . . . , aπn ) where (π1 , π2 , . . . , πn ) is one of the n! permutations of (1, 2, . . . , n). (a) The support function of C is deﬁned as SC (y ) = maxx∈C y T x. Show that SC (y ) = a[1] y[1] + a[2] y[2] + · · · + a[n] y[n] . (u[1] , u[2] , . . . , u[n] denote the components of an n-vector u in nonincreasing order.) Hint. To ﬁnd the maximum of y T x over x ∈ C , write the inner product as y...
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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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