bv_cvxbook_extra_exercises

# Then our original problem can be expressed as

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

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,...
View Full Document

## 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.

Ask a homework question - tutors are online