v2010.10.26 - Convex Optimization

5.4. EDM DEFINITION 429Crippen & Havel recommend working exclusively with distance data becausethey consider angle data to be mathematically cumbersome. The presentexample shows instead how inclusion of dihedral angle data into a problemstatement can be made elegant and convex.As before, ascribe position information to the matrixX = [x 1 · · · x N ] ∈ R 3×N (76)and introduce a matrix ℵ holding normals η to planes respecting dihedralangles ϕ :ℵ [η 1 · · · η M ] ∈ R 3×M (947)As in the other examples, we preferentially work with Gram matrices Gbecause of the bridge they provide between other variables; we define[ ] [ ] [ ]Gℵ Z ℵZ T T ℵ ℵ T X ℵT [ ℵ X ]G X X T ℵ X T =∈ RX X T N+M×N+M (948)whose rank is 3 by assumption. So our problem’s variables are the twoGram matrices G X and G ℵ and matrix Z = ℵ T X of cross products. Thenmeasurements of distance-square d can be expressed as linear constraints onG X as in (946), dihedral angle ϕ measurements can be expressed as linearconstraints on G ℵ by (919), and normal-vector η conditions can be expressedby vanishing linear constraints on cross-product matrix Z : Consider threepoints x labelled 1, 2, 3 assumed to lie in the l th plane whose normal is η l .There might occur, for example, the independent constraintsη T l (x 1 − x 2 ) = 0η T l (x 2 − x 3 ) = 0(949)which are expressible in terms of constant matrices A∈ R M×N ;〈Z , A l12 〉 = 0〈Z , A l23 〉 = 0(950)Although normals η can be constrained exactly to unit length,δ(G ℵ ) = 1 (951)NMR data is noisy; so measurements are given as upper and lower bounds.Given bounds on dihedral angles respecting 0 ≤ ϕ j ≤ π and bounds on