v2006.03.09 - Convex Optimization

164 CHAPTER 2. CONVEX GEOMETRY2.13.5.1 self-dual conesFrom (106) (a consequence of the halfspaces theorem (2.4.1.1.1)), where thesupport function for a convex cone always has 0 value, or from discretizeddefinition (296) of the dual cone we get a rather self-evident characterizationof self-duality:K = K ∗ ⇔ K = ⋂ {y | γ T y ≥ 0 } (300)γ∈G(K)In words: Cone K is self-dual iff its own extreme directions areinward-normals to a (minimal) set of hyperplanes bounding halfspaces whoseintersection constructs it. This means each extreme direction of K is normalto a hyperplane exposing one of its own faces; a necessary but insufficientcondition for self-duality (Figure 45, for example).Self-dual cones are of necessarily nonempty interior [22,I] and invariantto rotation about the origin. Their most prominent representatives are theorthants, the positive semidefinite cone S M + in the ambient space of symmetricmatrices (298), and the Lorentz cone (141) [15,II.A] [38, exmp.2.25]. Inthree dimensions, a plane containing the axis of revolution of a self-dual cone(and the origin) will produce a slice whose boundary makes a right angle.2.13.5.1.1 Example. Linear matrix inequality.Consider a peculiar vertex-description for a closed convex cone defined overthe positive semidefinite cone (instead of the nonnegative orthant as indefinition (79)): for X ∈ S n given A j ∈ S n , j =1... m⎧⎡⎨K = ⎣⎩⎧⎡⎨= ⎣⎩〈A 1 , X〉.〈A m , X〉⎤ ⎫⎬⎦ | X ≽ 0⎭ ⊆ Rm⎤ ⎫svec(A 1 ) T⎬. ⎦svec X | X ≽ 0svec(A m ) T ⎭∆= {A svec X | X ≽ 0}(301)where A∈ R m×n(n+1)/2 , and where symmetric vectorization svec is definedin (46). K is indeed a convex cone because by (138)A svec X p1 , A svec X p2 ∈ K ⇒ A(ζ svec X p1 +ξ svec X p2 ) ∈ K for all ζ,ξ ≥ 0(302)