This preview shows page 1. Sign up to view the full content.
Unformatted text preview: using a relatively painless method, leveraging some composition rules and
known convexity of a few other functions.
(a) Explain why t − (1/t)uT u is a concave function on dom f . Hint. Use convexity of the quadratic
over linear function.
(b) From this, show that − log(t − (1/t)uT u) is a convex function on dom f .
(c) From this, show that f is convex. 2.4 A quadraticoverlinear composition theorem. Suppose that f : Rn → R is nonnegative and convex,
and g : Rn → R is positive and concave. Show that the function f 2 /g , with domain dom f ∩ dom g ,
is convex.
2.5 A perspective composition rule. [4] Let f : Rn → R be a convex function with f (0) ≤ 0.
(a) Show that the perspective tf (x/t), with domain {(x, t)  t > 0, x/t ∈ dom f }, is nonincreasing
as a function of t. 5 (b) Let g be concave and positive on its domain. Show that the function
dom h = {x ∈ dom g  x/g (x) ∈ dom f } h(x) = g (x)f (x/g (x)),
is convex.
(c) As an example, show that
h(x) = xT x
,
( n=1 xk )1/n
k dom h = Rn
++ is convex.
2.6 Perspect...
View Full
Document
 Fall '13
 F.Borrelli
 The Aeneid

Click to edit the document details