bv_cvxbook_extra_exercises

This problem can be expressed as maximize t subject

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Unformatted text preview: t the origin) contains C . (a) Show that the condition E ⊆ C is equivalent to aT Qai ≤ 1 for i = 1, . . . , p. i (b) The volume of E is proportional to (det Q)1/2 , so we can find the maximum volume ellipsoid E inside C by solving the convex problem minimize log det Q−1 subject to aT Qai ≤ 1, i i = 1. . . . , p. (22) The variable is the matrix Q ∈ Sn and the domain of the objective function is Sn . ++ Derive the Lagrange dual of problem (22). (c) Note that Slater’s condition for (22) holds (aT Qai < 1 for Q = ǫI and ǫ > 0 small enough), i so we have strong duality, and the KKT conditions are necessary and sufficient for optimality. What are the KKT conditions for (22)? Suppose Q is optimal. Use the KKT conditions to show that In other words C ⊆ √ x∈C =⇒ xT Q−1 x ≤ n. nE , which is the desired result. 7.2 Euclidean distance matrices. A matrix X ∈ Sn is a Euclidean distance matrix if its elements xij can be expressed as xij = pi − pj 2 , i, j = 1, . . . , n, 2 for some vectors p1 , . . . , p...
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