v2009.01.01 - Convex Optimization

5.4. EDM DEFINITION 377
Crippen & Havel recommend working exclusively with distance data
because they consider angle data to be mathematically cumbersome. The
present example shows instead how inclusion of dihedral angle data into a
problem statement can be made elegant and convex.
As before, ascribe position information to the matrix
X = [x 1 · · · x N ] ∈ R 3×N (68)
and introduce a matrix ℵ holding normals η to planes respecting dihedral
angles ϕ :
ℵ ∆ = [η 1 · · · η M ] ∈ R 3×M (852)
As in the other examples, we preferentially work with Gram matrices G
because of the bridge they provide between other variables; we define
[ ] [ ] [ ]
Gℵ Z ∆ ℵ
Z T =
T ℵ ℵ T X ℵ
T [ ℵ X ]
G X X T ℵ X T =
∈ R
X X T N+M×N+M (853)
whose rank is 3 by assumption. So our problem’s variables are the two
Gram matrices G X and G ℵ and matrix Z = ℵ T X of cross products. Then
measurements of distance-square d can be expressed as linear constraints on
G X as in (851), dihedral angle ϕ measurements can be expressed as linear
constraints on G ℵ by (827), and normal-vector η conditions can be expressed
by vanishing linear constraints on cross-product matrix Z : Consider three
points 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 constraints
η T l (x 1 − x 2 ) = 0
η T l (x 2 − x 3 ) = 0
which are expressible in terms of constant matrices A∈ R M×N ;
〈Z , A l12 〉 = 0
〈Z , A l23 〉 = 0
Although normals η can be constrained exactly to unit length,
(854)
(855)
δ(G ℵ ) = 1 (856)
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
distances d i and given constant matrices A k (855) and symmetric matrices
Φ i (796) and B j per (827), then a molecular conformation problem can be
expressed: