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)
12th Edition
ISBN:9781305652231
Author:R. David Gustafson, Jeff Hughes
Publisher:R. David Gustafson, Jeff Hughes
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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