v2010.10.26 - Convex Optimization

5.11. EDM INDEFINITENESS 475holds only those rows of B corresponding to conically independent rowsin BV N .For presorted eigenvalues, (1107) can be equivalently restatedD ∈ EDM N⇔⎧⎪⎨⎪⎩([ 0 1Tλ1 −DD ∈ S N h])∈[ ]RN+∩ ∂HR −(1125)Vertex-description of the dual spectral cone is, (314)([ ]) 0 1T ∗λ1 −EDM N =[ ]RN++ ∂H ∗ ⊆ R N+1R −= {[ B T 1 −1 ] b | b ≽ 0 } =From (445) we get a halfspace-description:{[ }˜BT 1 −1]a | a ≽ 0(1126)([ ]) 0 1T ∗λ1 −EDM N = {y ∈ R N+1 | (VN T ˜B T ) † VN Ty ≽ 0}= {y ∈ R N+1 | [I −1 ]y ≽ 0}(1127)This polyhedral dual spectral cone is closed, convex, full-dimensional but isnot pointed. (Notice that any nonincreasingly ordered eigenspectrum belongsto this dual spectral cone.)5.11.2.4 Dual cone versus dual spectral coneAn open question regards the relationship of convex cones and their duals tothe corresponding spectral cones and their duals. A positive semidefinitecone, for example, is selfdual. Both the nonnegative orthant and themonotone nonnegative cone are spectral cones for it. When we considerthe nonnegative orthant, then that spectral cone for the selfdual positivesemidefinite cone is also selfdual.