additional_exercises_sol-homework_219-220.docx - Solve the...

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1 Solve the relaxed A -optimal experiment design problem, minimize (1 /m ) tr p λ v v T _1 subject to 1 T λ = 1 , λ 0 , with variable λ e R p . Find the optimal point λ and the associated optimal value of the relaxed problem. This optimal value is a lower bound on the optimal value of the discrete A -optimal experiment design problem, minimize tr p m v v T _1 subject to m 1 + . . . + m p = m, m i e { 0 , . . . , m } , i = 1 , . . . , p, with variables m 1 , . . . , m p . To get a suboptimal point for this discrete problem, round the entries in to obtain in tegers m ˆ i . If needed, adju s t these b y hand or some other method to ensure that they sum to m , and compute the objective value obtained. This is, of course, an upper bound on the optimal value of the discrete problem. Give the gap between this upper bound and the lower bound obtained from the relaxed problem. Note that the two objective values can be interpreted as mean-square estimation error E | x ˆ _ x | 2 . Solution. The objective of the relaxed problem is convex, so it is a convex problem.

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