Suppose f rn r is a posynomial function x y rn 1 and

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Unformatted text preview: r ℓp -norms with rational p. (a) Use the observation at the beginning of exercise 4.26 in Convex Optimization to express the constraint √ y, z1 , z2 ≥ 0, y ≤ z1 z2 , with variables y , z1 , z2 , as a second-order cone constraint. Then extend your result to the constraint y ≤ (z1 z2 · · · zn )1/n , y ≥ 0, z 0, where n is a positive integer, and the variables are y ∈ R and z ∈ Rn . First assume that n is a power of two, and then generalize your formulation to arbitrary positive integers. (b) Express the constraint f (x) ≤ t as a second-order cone constraint, for the following two convex functions f : xα x ≥ 0 0 x < 0, f (x) = where α is rational and greater than or equal to one, and f (x) = xα , dom f = R++ , where α is rational and negative. (c) Formulate the norm approximation problem minimize 22 Ax − b p as a second-order cone program, where p is a rational number greater than or equal to one. The variable in the optimization problem is x ∈ Rn . The matrix A ∈ Rm×n and the vector b ∈ Rm are...
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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 Berkeley.

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