v2009.01.01 - Convex Optimization

368 CHAPTER 5. EUCLIDEAN DISTANCE MATRIX
This preceding transformation of trilateration to a semidefinite program
works all the time ((845) holds) despite relaxation (842) because the optimal
solution set is a unique point.
5.4.2.2.5 Proof (sketch). Only the sensor location x 1 is unknown.
The objective function together with the equality constraints make a linear
system of equations in Gram matrix variable G
trG = ‖x 1 ‖ 2 + ‖ˇx 2 ‖ 2 + ‖ˇx 3 ‖ 2 + ‖ˇx 4 ‖ 2
tr(GΦ i1 = ďi1 , i = 2, 3, 4
tr ( ) (846)
Ge i e T i = ‖ˇxi ‖ 2 , i = 2, 3, 4
tr(G(e i e T j + e j e T i )/2) = ˇx T i ˇx j , 2 ≤ i < j = 3, 4
which is invertible:
⎡
⎤
svec(I) T
svec(Φ 21 ) T
svec(Φ 31 ) T
svec(Φ 41 ) T
svec(e 2 e T 2 ) T
svec G =
svec(e 3 e T 3 ) T
svec(e 4 e T 4 ) T
svec ( (e
⎢ 2 e T 3 + e 3 e T 2 )/2 ) T
⎣ svec ( (e 2 e T 4 + e 4 e T 2 )/2 ) T ⎥
svec ( (e 3 e T 4 + e 4 e T 3 )/2 ) ⎦
T
−1⎡
‖x 1 ‖ 2 + ‖ˇx 2 ‖ 2 + ‖ˇx 3 ‖ 2 + ‖ˇx 4 ‖ 2 ⎤
ď 21
ď 31
ď 41
‖ˇx 2 ‖ 2
‖ˇx 3 ‖ 2
‖ˇx 4 ‖ 2
⎢ ˇx T 2ˇx 3
⎥
⎣ ˇx T 2ˇx 4
⎦
ˇx T 3ˇx 4
(847)
That line in the ambient space S 4 of G , claimed on page 367, is traced by
‖x 1 ‖ 2 ∈ R on the right-hand side, as it turns out. One must show this line
to be tangential (2.1.7.2) to S 4 + in order to prove uniqueness. Tangency is
possible for affine dimension 1 or 2 while its occurrence depends completely
on the known measurement data.
But as soon as significant noise is introduced or whenever distance data is
incomplete, such problems can remain convex although the set of all optimal
solutions generally becomes a convex set bigger than a single point (and still
containing the noiseless solution).