16 polar of a set the polar of c rn is dened as the

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: result: If K is a closed convex cone, then K ∗∗ = K . Next, show that C ∩ D ⊇ (C ∗ + D∗ )∗ and conclude (C ∩ D)∗ = C ∗ + D∗ . (d) Show that the dual of the polyhedral cone V = {x | Ax V ∗ = {AT v | v 0} can be expressed as 0}. 1.6 Polar of a set. The polar of C ⊆ Rn is defined as the set C ◦ = {y ∈ Rn | y T x ≤ 1 for all x ∈ C }. (a) Show that C ◦ is convex (even if C is not). (b) What is the polar of a cone? (c) What is the polar of the unit ball for a norm ·? (d) Show that if C is closed and convex, with 0 ∈ int C , then (C ◦ )◦ = C . 4 2 Convex functions 2.1 Maximum of a convex function over a polyhedron. Show that the maximum of a convex function f over the polyhedron P = conv{v1 , . . . , vk } is achieved at one of its vertices, i.e., sup f (x) = max f (vi ). x∈P i=1,...,k (A stronger statement is: the maximum of a convex function over a closed bound convex set is achieved at an extreme point, i.e., a point in the set that is not a convex combination of any other points in the set.) Hint. Assume the statement is false, and use Jensen’s inequality. 2.2 A general vector composition rule...
View Full Document

Ask a homework question - tutors are online