bv_cvxbook_extra_exercises

x y 2 x y 2 52 minimax rational t to the

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Unformatted text preview: = 1T d = 1. Consider the geometric program minimize xT Ay n subject to i=1 n j =1 x ci = 1 i 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 matrix 1 B= T diag(x)A diag(y ) x Ay 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. (19) 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 minimize t subject to x − bi 2 2 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.

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