3.21 Pointwise maximum and supremum. Show that the following functions f: R" → R are convex. (a) f(x) = maxi=1,...,k ||A®x – b()||, where A@ E RmX", b(^) E R" and || || is a norm on Rm. (b) f(x) = E læ|ta on R", where |x| denotes the vector with |x|i = |xi| (i.e., |x| is the absolute value of x, componentwise), and |æ|1 is the ith largest component of |x|. In other words, |x||1], |x|12], ..., |x|[n] are the absolute values of the components of x, sorted in nonincreasing order. %3D
3.21 Pointwise maximum and supremum. Show that the following functions f: R" → R are convex. (a) f(x) = maxi=1,...,k ||A®x – b()||, where A@ E RmX", b(^) E R" and || || is a norm on Rm. (b) f(x) = E læ|ta on R", where |x| denotes the vector with |x|i = |xi| (i.e., |x| is the absolute value of x, componentwise), and |æ|1 is the ith largest component of |x|. In other words, |x||1], |x|12], ..., |x|[n] are the absolute values of the components of x, sorted in nonincreasing order. %3D
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
Section: Chapter Questions
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![3.21 Pointuwise maximum and supremum. Show that the following functions f : R" → R are
convex.
(a) f(x):
= max;=1,.,k ||Aª)x – b@)||, where A) e Rmxn, b(i) e R™ and || · || is a norm
on R™.
(b) ƒ(x) = E, a|g on R", where |æ| denotes the vector with |x|; = |xi| (i.e., |x| is
the absolute value of x, componentwise), and |x|f2] is the ith largest component of
|x|. In other words, |æ|1], |x|(2), ..., |æ|[n] are the absolute values of the components
of x, sorted in nonincreasing order.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe8fae079-28af-4fb7-b5d6-8bc2e6b2edbf%2Fa89e6e2d-4ce2-420e-a154-48f5ea424995%2Fzedn1i_processed.jpeg&w=3840&q=75)
Transcribed Image Text:3.21 Pointuwise maximum and supremum. Show that the following functions f : R" → R are
convex.
(a) f(x):
= max;=1,.,k ||Aª)x – b@)||, where A) e Rmxn, b(i) e R™ and || · || is a norm
on R™.
(b) ƒ(x) = E, a|g on R", where |æ| denotes the vector with |x|; = |xi| (i.e., |x| is
the absolute value of x, componentwise), and |x|f2] is the ith largest component of
|x|. In other words, |æ|1], |x|(2), ..., |æ|[n] are the absolute values of the components
of x, sorted in nonincreasing order.
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