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Unformatted text preview: given. For an mvector 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 onedimensional 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 θweighted 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.
 Fall '13
 F.Borrelli
 The Aeneid

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