v2010.10.26 - Convex Optimization

502 CHAPTER 6. CONE OF DISTANCE MATRICESN = 4. Relative-angle inequality (1162) together with four Euclidean metricproperties are necessary and sufficient tests for realizability oftetrahedra. (1163) Albeit relative angles θ ikj (954) are nonlinearfunctions of the d ij , relative-angle inequality provides a regulartetrahedron in R 3 [sic] (Figure 136) bounding angles θ ikj at vertexx k consistently with EDM 4 . 6.2Yet were we to employ the procedure outlined in5.14.3 for makinggeneralized triangle inequalities, then we would find all the necessary andsufficient d ij -transformations for generating bounding polyhedra consistentwith EDMs of any higher dimension (N > 3).6.3 √ EDM cone is not convexFor some applications, like a molecular conformation problem (Figure 3,Figure 127) or multidimensional scaling [103] [354], absolute distance √ d ijis the preferred variable. Taking square root of the entries in all EDMs Dof dimension N , we get another cone but not a convex cone when N > 3(Figure 138b): [89,4.5.2]√EDM N { ◦√ D | D ∈ EDM N } (1182)where ◦√ D is defined like (1181). It is a cone simply because any coneis completely constituted by rays emanating from the origin: (2.7) Anygiven ray {ζΓ∈ R N(N−1)/2 | ζ ≥0} remains a ray under entrywise square root:{ √ ζΓ∈ R N(N−1)/2 | ζ ≥0}. It is already established thatD ∈ EDM N ⇒ ◦√ D ∈ EDM N (1094)But because of how √ EDM N is defined, it is obvious that (confer5.10)D ∈ EDM N ⇔ ◦√ √D ∈ EDM N (1183)Were √ EDM N convex, then given◦ √ D 1 , ◦√ D 2 ∈ √ EDM N we wouldexpect their conic combination ◦√ D 1 + ◦√ D 2 to be a member of √ EDM N .6.2 Still, property-4 triangle inequalities (1062) corresponding to each principal 3 ×3submatrix of −VN TDV N demand that the corresponding √ d ij belong to a polyhedralcone like that in Figure 138b.