v2009.01.01 - Convex Optimization

378 CHAPTER 5. EUCLIDEAN DISTANCE MATRIX
find
G ℵ ∈S M , G X ∈S N , Z∈R M×N G X
subject to d i ≤ tr(G X Φ i ) ≤ d i , ∀i ∈ I 1
cos ϕ j ≤ tr(G ℵ B j ) ≤ cos ϕ j , ∀j ∈ I 2
〈Z , A k 〉 = 0 , ∀k ∈ I 3
G X 1 = 0
δ(G ℵ ) = 1
[ ]
Gℵ Z
Z T ≽ 0
G X
[ ]
Gℵ Z
rank
Z T = 3
G X
(857)
where G X 1=0 provides a geometrically centered list X (5.4.2.2). Ignoring
the rank constraint would tend to force cross-product matrix Z to zero.
What binds these three variables is the rank constraint; we show how to
satisfy it in4.4.
5.4.3 Inner-product form EDM definition
[p.20] We might, for example, realize a constellation given only
interstellar distance (or, equivalently, distance from Earth and
relative angular measurement; the Earth as vertex to two stars).
Equivalent to (794) is [332,1-7] [287,3.2]
d ij = d ik + d kj − 2 √ d ik d kj cos θ ikj
= [√ d ik
√
dkj
] [ 1 −e ıθ ikj
−e −ıθ ikj
1
] [√ ]
d ik
√
dkj
(858)
called the law of cosines, where ı ∆ = √ −1 , i,k,j are positive integers, and
θ 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 ‖
(859)