bv_cvxbook_extra_exercises

E x 1n1 and also with a pure investment in each asset

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Unformatted text preview: how that Tn (x1 , . . . , xn ) ≻ 0 for every feasible x in the SDP above. You can do this by induction on n. • For n = 1, the constraint is x1 ≥ 1 which obviously implies x1 > 0. • In the induction step, assume n ≥ 2 and that Tn−1 (x1 , . . . , xn−1 ) ≻ 0. Use a Schur complement argument and the Toeplitz structure of Tn to show that Tn (x1 , . . . , xn ) e1 eT 1 implies Tn (x1 , . . . , xn ) ≻ 0. (c) Suppose the optimal value of the SDP above is finite and attained, and that Z is dual optimal. Use the result of part (b) to show that the rank of Z is at most one, i.e., Z can be expressed as Z = yy T for some n-vector y . Show that y satisfies 2 2 2 y1 + y2 + · · · + yn = c 1 y1 y2 + y2 y3 + · · · + yn−1 yn = c2 /2 . . . y1 yn−1 + y2 yn = cn−1 /2 y1 yn = cn /2. This can be expressed as an identity |Y (ω )|2 = R(ω ) between two functions Y (ω ) = y1 + y2 e−iω + y3 e−3iω + · · · + yn e−i(n−1)ω (with i = R(ω ). √ R(ω ) = c1 + c2 cos ω + c3 cos(2ω ) + · · · + cn cos((n...
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