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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. (1)

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. (1)

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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please send handwritten solution for 3.20 part a

 

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.)
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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