v2007.09.17 - Convex Optimization

324 CHAPTER 5. EUCLIDEAN DISTANCE MATRIXThen we can express the molecular conformation problem: for 0 ≤ ϕ ≤ πand constant symmetric matrices BfindG 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 3δ(G ℵ ) = 1[ ]Gℵ ZZ T ≽ 0G X[ ]Gℵ ZrankZ T = 3G X(768)Ignoring the rank constraint tends to force inner-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 definition[p.20] We might, for example, realize a constellation given onlyinterstellar distance (or, equivalently, distance from Earth andrelative angular measurement; the Earth as vertex to two stars).Equivalent to (705) is [288,1-7] [249,3.2]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(769)called the law of cosines, where ı ∆ = √ −1 , i,k,j are positive integers, andθ ikj is 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 ‖(770)