bv_cvxbook_extra_exercises

C from this show that f is convex 24 a quadratic over

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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 quadratic-over-linear 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

Ask a homework question - tutors are online