bv_cvxbook_extra_exercises

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

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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...
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