Then our original problem can be expressed as

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Unformatted text preview: )−1 , with domain Sn × Sn , is concave. ++ ++ Hint. Verify that the matrix R= A−1 I B −1 − I is nonsingular. Then apply the congruence transformation defined by R to the two sides of matrix inequality in the SDP, to obtain an equivalent inequality RT XX XX RT R A0 0B R. 3.14 Trace of geometric mean of matrices. [1] G(A, B ) = A1/2 A−1/2 BA−1/2 1/2 A1/2 is known as the geometric mean of positive definite matrices A and B . Show that X = G(A, B ) solves the SDP maximize tr X AX subject to 0. XB The variable is X ∈ Sn . The matrices A ∈ Sn and B ∈ Sn are given. ++ ++ Conclude that the function tr G(A, B ) is concave, for A, B positive definite. Hint. The symmetric matrix square root is monotone: if U and V are positive semidefinite with U V then U 1/2 V 1/2 . 3.15 Transforming a standard form convex problem to conic form. In this problem we show that any convex problem can be cast in conic form, provided some technical conditions hold. We start with a standard form convex problem with linear objective (without loss of generality): minimize cT x subject to fi (x) ≤ 0, Ax = b, 17 i = 1, . . . , m, where fi : Rn → R are convex,...
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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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