v2007.09.13 - Convex Optimization

6.8. DUAL EDM CONE 419When the Finsler criterion (1047) is applied despite lower affinedimension, the constant κ can go to infinity making the test −D+κ11 T ≽ 0impractical for numerical computation. Chabrillac & Crouzeix invent acriterion for the semidefinite case, but is no more practical: for D ∈ S N hD ∈ EDM N⇔(1048)∃κ p >0 ∀κ≥κ p , −D − κ11 T [sic] has exactly one negative eigenvalue6.8 Dual EDM cone6.8.1 Ambient S NWe consider finding the ordinary dual EDM cone in ambient space S N whereEDM N is pointed, closed, convex, but has empty interior. The set of all EDMsin S N is a closed convex cone because it is the intersection of halfspaces aboutthe 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(1049)By definition (258), dual cone K ∗comprises each and every vectorinward-normal to a hyperplane supporting convex cone K (2.4.2.6.1) orbounding a halfspace containing K . The dual EDM cone in the ambientspace of symmetric matrices is therefore expressible as the aggregate of everyconic combination of inward-normals from (1049):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=1j=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+ } (1050)The EDM cone is not self-dual in ambient S N because its affine hull belongsto a proper subspaceaff EDM N = S N h (1051)