v2007.09.17 - Convex Optimization

110 CHAPTER 2. CONVEX GEOMETRYA conic section is the intersection of a cone with any hyperplane. In threedimensions, an intersecting plane perpendicular to a circular cone’s axis ofrevolution produces a section bounded by a circle. (Figure 34) A prominentexample of a circular cone in convex analysis is the Lorentz cone (147). Wealso find that the positive semidefinite cone and cone of Euclidean distancematrices are circular cones, but only in low dimension.The positive semidefinite cone has axis of revolution that is the ray(base 0) through the identity matrix I . Consider the set of normalizedextreme directions of the positive semidefinite cone: for some arbitrarypositive constant a∈ R +{yy T ∈ S M | ‖y‖ = √ a} ⊂ ∂S M + (200)The distance from each extreme direction to the axis of revolution is theradius√R ∆ = infc ‖yyT − cI‖ F = a1 − 1 M(201)which is the distance from yy T to a M I ; the length of vector yyT − a M I .Because distance R (in a particular dimension) from the axis of revolutionto each and every normalized extreme direction is identical, the extremedirections lie on the boundary of a hypersphere in isometrically isomorphicR M(M+1)/2 . From Example 2.9.2.4.1, the convex hull (excluding the vertexat the origin) of the normalized extreme directions is a conic sectionC ∆ = conv{yy T | y ∈ R M , y T y = a} = S M + ∩ {A∈ S M | 〈I , A〉 = a} (202)orthogonal to the identity matrix I ;〈C − a M I , I〉 = tr(C − a I) = 0 (203)MAlthough the positive semidefinite cone possesses some characteristics ofa circular cone, we can prove it is not by demonstrating a shortage of extremedirections; id est, some extreme directions corresponding to each and everyangle of rotation about the axis of revolution are nonexistent: Referring toFigure 35, [288,1-7]