3.21 Pointwise maximum and supremum. Show that the following functions f: R" → R are convex. (a) f(x) = max;=1,..,k ||A®) x - b)||, where A@) E Rmxn, b) E Rm and || · || is a norm on Rm. (b) f(x) = E-1 læ|t] on R", where |a| denotes the vector with |æ|i = |xi| (i.e., |a| is the absolute value of x, componentwise), and |1] is the ith largest component of |x|. In other words, ||1), |x|12), ..., |æ|[n] are the absolute values of the components of x, sorted in nonincreasing order.
3.21 Pointwise maximum and supremum. Show that the following functions f: R" → R are convex. (a) f(x) = max;=1,..,k ||A®) x - b)||, where A@) E Rmxn, b) E Rm and || · || is a norm on Rm. (b) f(x) = E-1 læ|t] on R", where |a| denotes the vector with |æ|i = |xi| (i.e., |a| is the absolute value of x, componentwise), and |1] is the ith largest component of |x|. In other words, ||1), |x|12), ..., |æ|[n] are the absolute values of the components of x, sorted in nonincreasing order.
College Algebra (MindTap Course List)
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Chapter3: Functions
Section3.3: More On Functions; Piecewise-defined Functions
Problem 99E: Determine if the statemment is true or false. If the statement is false, then correct it and make it...
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please send handwritten solution for Q 3.21 part b
Only Handwritten solution accepted
![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%2F3dea7229-6f7c-4b79-9794-24c72fc8aed1%2Fu39dyri_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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