v2010.10.26 - Convex Optimization

182 CHAPTER 2. CONVEX GEOMETRYK = K ∗ ⇔ K = ⋂ {y | γ T y ≥ 0 } (379)γ∈G(K)In words: Cone K is selfdual iff its own extreme directions are inward-normalsto a (minimal) set of hyperplanes bounding halfspaces whose intersectionconstructs it. This means each extreme direction of K is normal to ahyperplane exposing one of its own faces; a necessary but insufficientcondition for selfdualness (Figure 62, for example).Selfdual cones are necessarily full-dimensional. [30,I] Their mostprominent representatives are the orthants (Cartesian cones), the positivesemidefinite cone S M + in the ambient space of symmetric matrices (377), andLorentz cone (178) [20,II.A] [61, exmp.2.25]. In three dimensions, a planecontaining the axis of revolution of a selfdual cone (and the origin) willproduce a slice whose boundary makes a right angle.2.13.5.1.1 Example. Linear matrix inequality. (confer2.13.2.0.3)Consider a peculiar vertex-description for a convex cone K defined overa positive semidefinite cone (instead of a nonnegative orthant as indefinition (103)): 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}(380)where A∈ R m×n(n+1)/2 , and where symmetric vectorization svec is definedin (56). Cone K is indeed convex because, by (175)A svec X p1 , A svec X p2 ∈ K ⇒ A(ζ svec X p1 +ξ svec X p2 )∈ K for all ζ,ξ ≥ 0(381)since a nonnegatively weighted sum of positive semidefinite matrices must bepositive semidefinite. (A.3.1.0.2) Although matrix A is finite-dimensional, Kis generally not a polyhedral cone (unless m=1 or 2) simply because X ∈ S n + .