bv_cvxbook_extra_exercises

There is nothing special about n 4 here the same

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Unformatted text preview: + dom f = Rn ++ k=1 is concave, where αk are nonnegative numbers with k αk = 1. 2.11 Suppose that f : Rn → R is convex, and define g (x, t) = f (x/t), dom g = {(x, t) | x/t ∈ dom f, t > 0}. Show that g is quasiconvex. 6 2.12 Continued fraction function. Show that the function 1 f (x) = x1 − 1 x2 − 1 x3 − 1 x4 defined where every denominator is positive, is convex and decreasing. (There is nothing special about n = 4 here; the same holds for any number of variables.) 2.13 Circularly symmetric Huber function. The scalar Huber function is defined as fhub (x) = (1/2)x2 | x| ≤ 1 |x| − 1/2 |x| > 1. This convex function comes up in several applications, including robust estimation. This problem concerns generalizations of the Huber function to Rn . One generalization to Rn is given by fhub (x1 ) + · · · + fhub (xn ), but this function is not circularly symmetric, i.e., invariant under transformation of x by an orthogonal matrix. A generalization to Rn that is circularly symmetric is (1/2) x 2...
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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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