bv_cvxbook_extra_exercises

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Unformatted text preview: n how to solve the following two problems using convex optimization. Your solution can involve solving multiple convex problems, as long as the number of such problems is no more than linear in the dimensions n, k, p. (a) How would you determine whether ARk ⊆ B Rp ? This means that every nonnegative linear + + combination of the columns of A can be expressed as a nonnegative linear combination of the columns of B . (b) How would you determine whether ARk = Rn ? This means that every vector in Rn can be + expressed as a nonnegative linear combination of the columns of A. 7.19 Projection on convex hull of union of ellipsoids. Let E1 , . . . , Em be m ellipsoids in Rn defined as Ei = {Ai u + bi | u 2 ≤ 1}, i = 1, . . . , m, with Ai ∈ Rn×n and bi ∈ Rn . Consider the problem of projecting a point a ∈ Rn on the convex hull of the union of the ellipsoids: minimize x−a 2 subject to x ∈ conv(E1 ∪ · · · ∪ Em ). Formulate this as a second order cone program. 7.20 Bregman divergences. Let f : Rn → R be strictly convex and differen...
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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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