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Unformatted text preview: given. For an m-vector y , the norm y p is defined as 1/p m y p = k=1 | yk | p when p ≥ 1. 3.22 Linear optimization over the complement of a convex set. Suppose C ⊆ Rn is a closed bounded + convex set with 0 ∈ C , and c ∈ Rn . We define + ˜ C = cl(Rn \ C ) = cl{x ∈ Rn | x ∈ C}, + + which is the closure of the complement of C in Rn . + T x has a minimizer over C of the form αe , where α ≥ 0 and e is the k th standard ˜ Show that c k k unit vector. (If you have not had a course on analysis, you can give an intuitive argument.) ˜ If follows that we can minimize cT x over C by solving n one-dimensional optimization problems (which, indeed, can each be solved by bisection, provided we can check whether a point is in C or not). 3.23 Jensen’s inequality for posynomials. Suppose f : Rn → R is a posynomial function, x, y ∈ Rn , ++ 1 and θ ∈ [0, 1]. Define z ∈ Rn by zi = xθ yi −θ , i = 1, . . . , n. Show that f (z ) ≤ f (x)θ f (y )1−θ . ++ i Interpretation. We can think of z as a θ-weighted geometric mean between x and y . So the statement above is that a posy...
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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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