v2007.09.13 - Convex Optimization

156 CHAPTER 2. CONVEX GEOMETRY2.13.5.1.1 Example. Linear matrix inequality. (confer2.13.2.0.3)Consider a peculiar vertex-description for a closed convex cone defined overthe positive semidefinite cone (instead of the nonnegative orthant as indefinition (83)): 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}(324)where A∈ R m×n(n+1)/2 , and where symmetric vectorization svec is definedin (47). K is indeed a convex cone because by (144)A svec X p1 , A svec X p2 ∈ K ⇒ A(ζ svec X p1 +ξ svec X p2 ) ∈ K for all ζ,ξ ≥ 0(325)since a nonnegatively weighted sum of positive semidefinite matrices must bepositive semidefinite. (A.3.1.0.2) Although matrix A is finite-dimensional,K is generally not a polyhedral cone (unless m equals 1 or 2) becauseX ∈ S n + . Provided the A j matrices are linearly independent, thenrel int K = int K (326)meaning, the cone interior is nonempty implying the dual cone is pointedby (268).If matrix A has no nullspace, on the other hand, then (by2.10.1.1 andDefinition 2.2.1.0.1) A svec X is an isomorphism in X between the positivesemidefinite cone and R(A). In that case, convex cone K has relativeinteriorrel int K = {A svec X | X ≻ 0} (327)and boundaryrel ∂K = {A svec X | X ≽ 0, X ⊁ 0} (328)