v2009.01.01 - Convex Optimization

6.8. DUAL EDM CONE 477
When the Finsler criterion (1150) is applied despite lower affine
dimension, the constant κ can go to infinity making the test −D+κ11 T ≽ 0
impractical for numerical computation. Chabrillac & Crouzeix invent a
criterion for the semidefinite case, but is no more practical: for D ∈ S N h
D ∈ EDM N
⇔
(1151)
∃κ p >0 ∀κ≥κ p , −D − κ11 T [sic] has exactly one negative eigenvalue
6.8 Dual EDM cone
6.8.1 Ambient S N
We consider finding the ordinary dual EDM cone in ambient space S N where
EDM N is pointed, closed, convex, but has empty interior. The set of all EDMs
in S N is a closed convex cone because it is the intersection of halfspaces about
the origin in vectorized variable D (2.4.1.1.1,2.7.2):
EDM N = ⋂ {
D ∈ S N | 〈e i e T i , D〉 ≥ 0, 〈e i e T i , D〉 ≤ 0, 〈zz T , −D〉 ≥ 0 }
z∈ N(1 T )
i=1...N
(1152)
By definition (270), dual cone K ∗
comprises each and every vector
inward-normal to a hyperplane supporting convex cone K (2.4.2.6.1) or
bounding a halfspace containing K . The dual EDM cone in the ambient
space of symmetric matrices is therefore expressible as the aggregate of every
conic combination of inward-normals from (1152):
EDM N∗ = cone{e i e T i , −e j e T j | i, j =1... N } − cone{zz T | 11 T zz T =0}
∑
= { N ∑
ζ i e i e T i − N ξ j e j e T j | ζ i ,ξ j ≥ 0} − cone{zz T | 11 T zz T =0}
i=1
j=1
= {δ(u) | u∈ R N } − cone { V N υυ T V T N | υ ∈ RN−1 , (‖v‖= 1) } ⊂ S N
= {δ 2 (Y ) − V N ΨV T N | Y ∈ SN , Ψ∈ S N−1
+ } (1153)
The EDM cone is not self-dual in ambient S N because its affine hull belongs
to a proper subspace
aff EDM N = S N h (1154)