v2009.01.01 - Convex Optimization

106 CHAPTER 2. CONVEX GEOMETRY
γ
svec ∂ S 2 +
[ α β
β γ
]
α
√
2β
Minimal set of generators are the extreme directions: svec{yy T | y ∈ R M }
Figure 37: Truncated boundary of PSD cone in S 2 plotted in isometrically
isomorphic R 3 via svec (49); courtesy, Alexandre W. d’Aspremont. (Plotted
is 0-contour of smallest eigenvalue (175). Lightest shading is closest.
Darkest shading is furthest and inside shell.) Entire boundary can be
constructed from an aggregate of rays (2.7.0.0.1) emanating exclusively
from the origin, { √
κ 2 [z1
2 2z1 z 2 z2 2 ] T | κ∈ R } . In this dimension the cone
is circular (2.9.2.5) while each and every ray on boundary corresponds to
an extreme direction, but such is not the case in any higher dimension
(confer Figure 20). PSD cone geometry is not as simple in higher dimensions
[25,II.12], although for real matrices it is self-dual (334) in ambient space
of symmetric matrices. [170,II] PSD cone has no two-dimensional faces in
any dimension, and its only extreme point resides at the origin.