3.20 Composition with an affine function. Show that the following functions f: R" -→R are convex. (a) f(x) = || Ax – b||, where A E Rmxn, b E R", and || · || is a norm on Rm. (b) f(x) = - (det(Ao + x1A1 + ..+xn An))'/m, on {x | Ao +x1A1+ +xn An > 0}, where Ai E S™. (c) f(X)= tr (Ao + ¤1A1 + + xn An), on {x | Ao+x1A1+.+xnAn > 0}, where A¡ E Sm. (Use the fact that tr(X-) is convex on S; see exercise 3.18.)
3.20 Composition with an affine function. Show that the following functions f: R" -→R are convex. (a) f(x) = || Ax – b||, where A E Rmxn, b E R", and || · || is a norm on Rm. (b) f(x) = - (det(Ao + x1A1 + ..+xn An))'/m, on {x | Ao +x1A1+ +xn An > 0}, where Ai E S™. (c) f(X)= tr (Ao + ¤1A1 + + xn An), on {x | Ao+x1A1+.+xnAn > 0}, where A¡ E Sm. (Use the fact that tr(X-) is convex on S; see exercise 3.18.)
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter6: Vector Spaces
Section6.5: The Kernel And Range Of A Linear Transformation
Problem 30EQ
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Transcribed Image Text:3.20 Composition with an affine function. Show that the following functions f : R" –→ R are
convex.
(a) f(x) = || Ax – b||, where A E RmXn, b E R™, and || · || is a norm on Rm.
(b) f(x) = – (det(Ao + ¤1A1 + ...+xn An))'/m, on {x | Ao + x1A1+ · ·+ xn An > 0},
where Ai E S™.
(c) f(X)= tr (Ao + ¤1A1 + · .+ xnAn)¬, on {x | Ao+¤1A1+…+xnAn > 0}, where
Ai E Sm. (Use the fact that tr(X-') is convex on ST+; see exercise 3.18.)
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