v2010.10.26 - Convex Optimization

6.8. DUAL EDM CONE 529to the geometric center subspace S N c (implicit constraint G1 = 0) keeps Gindependent of its translation-invariant subspace S N⊥c (5.5.1.1, Figure 148)so as not to become numerically unbounded.A problem dual to maximum variance unfolding problem (1246) (less theGram rank constraint) has been called the fastest mixing Markov process.That dual has simple interpretations in graph and circuit theory and inmechanical and thermal systems, explored in [340], and has direct applicationto quick calculation of PageRank by search engines [231,4]. Optimal Gramrank turns out to be tightly bounded above by minimum multiplicity of thesecond smallest eigenvalue of a dual optimal variable.6.8 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 not full-dimensional. The set of allEDMs in S N is a closed convex cone because it is the intersection of halfspacesabout 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(1247)By definition (297), dual cone K ∗comprises each and every vectorinward-normal to a hyperplane supporting convex cone K . The dual EDMcone in the ambient space of symmetric matrices is therefore expressible asthe aggregate of every conic combination of inward-normals from (1247):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+ } (1248)The EDM cone is not selfdual in ambient S N because its affine hull belongsto a proper subspaceaff EDM N = S N h (1249)