v2006.03.09 - Convex Optimization

4.11. EDM INDEFINITENESS 285For unordered eigenvalues, (654) can be equivalently restatedD ∈ EDM N⇔⎧⎪⎨⎪⎩([ 0 1Tλ1 −DD ∈ S N h])∈[ ]RN+∩ ∂HR −(672)Vertex-description of the dual spectral cone is, (257)([ ]) 0 1T ∗λ1 −EDM N =[ ]RN++ ∂H ∗ ⊆ R N+1R −= {[ B T 1 −1 ] b | b ≽ 0 } =From (363) we get a halfspace-description:{ [ }˜BT 1 −1]a | a ≽ 0(673)([ ]) 0 1T ∗λ1 −EDM N = {y ∈ R N+1 | (VN T ˜B T ) † VN T y ≽ 0}= {y ∈ R N+1 | [I −1 ]y ≽ 0}(674)This polyhedral dual spectral cone is closed, convex, has nonempty interiorbut is not pointed. (Notice that any nonincreasingly ordered eigenspectrumbelongs to this dual spectral cone.)4.11.2.4 Dual cone versus dual spectral coneAn open question regards the relationship of convex cones and their duals tothe corresponding spectral cones and their duals. The positive semidefinitecone, for example, is self-dual. Both the nonnegative orthant and themonotone nonnegative cone are spectral cones for it. When we considerthe nonnegative orthant, then the spectral cone for the self-dual positivesemidefinite cone is also self-dual.