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Unformatted text preview: = 1T d = 1. Consider the geometric
minimize xT Ay
n subject to i=1
j =1 x ci = 1
d yj j = 1, with variables x, y ∈ Rn (and implicit constraints x ≻ 0, y ≻ 0). Write this geometric program
in convex form and derive the optimality conditions. Show that if x and y are optimal, then the
diag(x)A diag(y )
satisﬁes B 1 = c and B T 1 = d.
4.23 A theorem due to Schoenberg. Suppose m balls in Rn , with centers ai and radii ri , have a nonempty
intersection. We deﬁne y to be a point in the intersection, so
y − ai 2 ≤ ri , i = 1, . . . , m. (18) Suppose we move the centers to new positions bi in such a way that the distances between the
centers do not increase:
bi − bj 2 ≤ ai − aj 2 , i, j = 1, . . . , m.
We will prove that the intersection of the translated balls is nonempty, i.e., there exists a point x
with x − bi 2 ≤ ri , i = 1, . . . , m. To show this we prove that the optimal value of
subject to x − bi 2
≤ ri + t, i = 1, . . . , m, (20) with variables x ∈ Rn and t ∈ R, is less t...
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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
- The Aeneid