v2010.10.26 - Convex Optimization

130 CHAPTER 2. CONVEX GEOMETRY2.9.2.8 Positive semidefinite cone is generally not circularExtreme angle equation (239) suggests that the positive semidefinite conemight be invariant to rotation about its axis of revolution; id est, a circularcone. We investigate this now:2.9.2.8.1 Definition. Circular cone: 2.47a pointed closed convex cone having hyperspherical sections orthogonal toits axis of revolution about which the cone is invariant to rotation. △A 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 46) A prominentexample of a circular cone in convex analysis is Lorentz cone (178). 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 a set of normalized extremedirections of the positive semidefinite cone: for some arbitrary positiveconstant a∈ R +{yy T ∈ S M | ‖y‖ = √ a} ⊂ ∂S M + (242)The distance from each extreme direction to the axis of revolution is radius√R infc ‖yyT − cI‖ F = a1 − 1 M(243)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 2.9.2.7.1, the convex hull (excluding vertex atthe 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} (244)orthogonal to identity matrix I ;〈C − a 〉M I , I = tr(C − a I) = 0 (245)M