v2007.09.13 - Convex Optimization

6.3.√EDM CONE IS NOT CONVEX 393N = 4. Relative-angle inequality (965) together with four Euclidean metricproperties are necessary and sufficient tests for realizability oftetrahedra. (966) Albeit relative angles θ ikj (766) are nonlinearfunctions of the d ij , relative-angle inequality provides a regulartetrahedron in R 3 [sic] (Figure 92) bounding angles θ ikj at vertex x kconsistently 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 the molecular conformation problem (Figure 3)or multidimensional scaling, [68] [265] absolute distance √ d ij is the preferredvariable. Taking square root of the entries in all EDMs D of dimension N ,we get another cone but not a convex cone when N > 3 (Figure 94(b)):[61,4.5.2]√EDM N ∆ = { ◦√ D | D ∈ EDM N } (985)where ◦√ D is defined as in (984). 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}. Because of how √ EDM N is defined, it is obviousthat (confer5.10)D ∈ EDM N ⇔ ◦√ √D ∈ EDM N (986)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 .That is easily proven false by counter-example via (986), for then( ◦√ D 1 + ◦√ D 2 ) ◦( ◦√ D 1 + ◦√ D 2 ) would need to be a member of EDM N .Notwithstanding, in7.2.1 we learn how to transform a nonconvexproximity problem in the natural coordinates √ d ij to a convex optimization.6.2 Still, property-4 triangle inequalities (866) corresponding to each principal 3 ×3submatrix of −VN TDV N demand that the corresponding √ d ij belong to a polyhedralcone like that in Figure 94(b).