v2009.01.01 - Convex Optimization

5.11. EDM INDEFINITENESS 423
where V N ∈ R N+1×N and
˜B ∆ =
⎡
⎢
⎣
e T 1
e T2 .
e T N
⎤
⎥
⎦ ∈ RN×N+1 (1029)
holds only those rows of B corresponding to conically independent rows
in BV N .
For presorted eigenvalues, (1012) can be equivalently restated
D ∈ EDM N
⇔
([ ]) 0 1
T ∗
λ
1 −EDM N =
⎧ ([ ])
⎪⎨ 0 1
T
λ
∈
1 −D
⎪⎩
D ∈ S N h
[ ]
R
N
+
∩ ∂H
R −
Vertex-description of the dual spectral cone is, (284)
[ ]
R
N
+
+ ∂H ∗ ⊆ R N+1
R −
= {[ B T 1 −1 ] b | b ≽ 0 } =
From (399) we get a halfspace-description:
(1030)
{[ }
˜BT 1 −1]
a | a ≽ 0
(1031)
([ ]) 0 1
T ∗
λ
1 −EDM N = {y ∈ R N+1 | (VN T ˜B T ) † VN Ty ≽ 0}
= {y ∈ R N+1 | [I −1 ]y ≽ 0}
(1032)
This polyhedral dual spectral cone is closed, convex, has nonempty interior
but is not pointed. (Notice that any nonincreasingly ordered eigenspectrum
belongs to this dual spectral cone.)
5.11.2.4 Dual cone versus dual spectral cone
An open question regards the relationship of convex cones and their duals to
the corresponding spectral cones and their duals. A positive semidefinite
cone, for example, is self-dual. Both the nonnegative orthant and the
monotone nonnegative cone are spectral cones for it. When we consider
the nonnegative orthant, then that spectral cone for the self-dual positive
semidefinite cone is also self-dual.