v2010.10.26 - Convex Optimization

430 CHAPTER 5. EUCLIDEAN DISTANCE MATRIXdistances d i and given constant matrices A k (950) and symmetric matricesΦ i (889) and B j per (919), then a molecular conformation problem can beexpressed:findG ℵ ∈S M , G X ∈S N , Z∈R M×N G Xsubject to d i ≤ tr(G X Φ i ) ≤ d i , ∀i ∈ I 1cos ϕ j ≤ tr(G ℵ B j ) ≤ cos ϕ j , ∀j ∈ I 2〈Z , A k 〉 = 0 , ∀k ∈ I 3G X 1 = 0δ(G ℵ ) = 1[ ]Gℵ ZZ T ≽ 0G X[ ]Gℵ ZrankZ T = 3G X(952)where G X 1=0 provides a geometrically centered list X (5.4.2.2). Ignoringthe rank constraint would tend to force cross-product matrix Z to zero.What binds these three variables is the rank constraint; we show how tosatisfy it in4.4.5.4.3 Inner-product form EDM definitionWe might, for example, want to realize a constellation given onlyinterstellar distance (or, equivalently, parsecs from our Sun andrelative angular measurement; the Sun as vertex to two distantstars); called stellar cartography.(p.22)Equivalent to (887) is [380,1-7] [331,3.2]d ij = d ik + d kj − 2 √ d ik d kj cos θ ikj= [√ d ik√dkj] [ 1 −e ıθ ikj−e −ıθ ikj1] [√ ]d ik√dkj(953)called law of cosines where ı √ −1 , i , j , k are positive integers, and θ ikjis the angle at vertex x k formed by vectors x i − x k and x j − x k ;cos θ ikj =1(d 2 ik + d kj − d ij )√ = (x i − x k ) T (x j − x k )dik d kj ‖x i − x k ‖ ‖x j − x k ‖(954)