v2010.10.26 - Convex Optimization

2.9. POSITIVE SEMIDEFINITE (PSD) CONE 131yy TaI θ aMRM 11TFigure 47: Illustrated is a section, perpendicular to axis of revolution, ofcircular cone from Figure 46. Radius R is distance from any extremedirection to axis at a I . Vector aM M 11T is an arbitrary reference by whichto measure angle θ .Proof. Although the positive semidefinite cone possesses somecharacteristics of a circular cone, we can show it is not by demonstratingshortage of extreme directions; id est, some extreme directions correspondingto each and every angle of rotation about the axis of revolution arenonexistent: Referring to Figure 47, [380,1-7]cosθ =〈 aM 11T − a M I , yyT − a M I〉a 2 (1 − 1 M ) (246)Solving for vector y we geta(1 + (M −1) cosθ) = (1 T y) 2 (247)which does not have real solution ∀ 0 ≤ θ ≤ 2π in every matrix dimension M .2.47 A circular cone is assumed convex throughout, although not so by other authors. Wealso assume a right circular cone.