bv_cvxbook_extra_exercises

Show that x ga b solves the sdp maximize tr x ax

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Unformatted text preview: f = {x ∈ Rn | F (x) ≻ 0}. (a) Minimize f (x) = cT F (x)−1 c where c ∈ Rm . (b) Minimize f (x) = maxi=1,...,K cT F (x)−1 ci where ci ∈ Rm , i = 1, . . . , K . i (c) Minimize f (x) = sup cT F (x)−1 c. c 2 ≤1 (d) Minimize f (x) = E(cT F (x)−1 c) where c is a random vector with mean E c = c and covariance ¯ E(c − c)(c − c)T = S . ¯ ¯ 16 3.12 A matrix fractional function.[1] Show that X = B T A−1 B solves the SDP minimize tr X A subject to BT B X 0, with variable X ∈ Sn , where A ∈ Sm and B ∈ Rm×n are given. ++ Conclude that tr(B T A−1 B ) is a convex function of (A, B ), for A positive definite. 3.13 Trace of harmonic mean of matrices. [1] The matrix H (A, B ) = 2(A−1 + B −1 )−1 is known as the harmonic mean of positive definite matrices A and B . Show that X = (1/2)H (A, B ) solves the SDP maximize tr X XX A0 subject to , XX 0B with variable X ∈ Sn . The matrices A ∈ Sn and B ∈ Sn are given. Conclude that the function ++ ++ tr (A−1 + B −1...
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