v2010.10.26 - Convex Optimization

2.9. POSITIVE SEMIDEFINITE (PSD) CONE 115γsvec ∂ S 2 +[ α ββ γ]α√2βMinimal set of generators are the extreme directions: svec{yy T | y ∈ R M }Figure 43: (d’Aspremont) Truncated boundary of PSD cone in S 2 plotted inisometrically isomorphic R 3 via svec (56); 0-contour of smallest eigenvalue(193). Lightest shading is closest, darkest shading is farthest and inside shell.Entire boundary can be constructed{ from an√ aggregate of rays (2.7.0.0.1)emanating exclusively from origin: κ 2 [z12 2z1 z 2 z2 2 ] T | κ∈ R , z ∈ R 2} .A circular cone in this dimension (2.9.2.8), each and every ray on boundarycorresponds to an extreme direction but such is not the case in any higherdimension (confer Figure 24). PSD cone geometry is not as simple in higherdimensions [26,II.12] although PSD cone is selfdual (377) in ambient realspace of symmetric matrices. [195,II] PSD cone has no two-dimensionalface in any dimension, its only extreme point residing at 0.