# HW3 - ELE 704 Optimization HW3 Due 15 March 2007 1 Prove...

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ELE 704 Optimization HW3 Due 15 March 2007 1. Prove the following: (a) If f and g are convex, both nondecreasing (or nonincreasing), and positive functions on an interval, then f ( g ( x )) is convex. (b) If f , g are concave, positive, with one nondecreasing and the other nonincreasing, then f ( g ( x )) is concave. (c) If f is convex, nondecreasing, and positive, and g is concave, nonin- creasing, and positive, then f/g is convex. 2. Does A - 1 exist and why, when (a) A 0 . (b) A ´ 0 . (c) A 0 . (d) A 0 . (e) What can be the connection between this problem and the update equation of Newton’s Algorithm. 3. The following is a common problem which you may encounter in diﬀerent problems. Let us denote the smallest and largest eigenvalues of a symmetric matrix M by λ min ( M ) and λ max ( M ) . Consider the program (a) minimize x T Mx s.t . k x k = 1 Prove that p * = λ min ( M ) . (b)

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## This note was uploaded on 05/25/2011 for the course ELECTRONIC 704 taught by Professor Cenktoker during the Spring '11 term at Hacettepe Üniversitesi.

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HW3 - ELE 704 Optimization HW3 Due 15 March 2007 1 Prove...

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