v2007.09.13 - Convex Optimization

312 CHAPTER 5. EUCLIDEAN DISTANCE MATRIXThis preceding transformation of trilateration to a semidefinite programworks all the time ((752) holds) despite relaxation (749) because the optimalsolution set is a unique point.Proof (sketch). Only the sensor location x 1 is unknown. The objectivefunction together with the equality constraints make a linear system ofequations in Gram matrix variable GtrG = ‖x 1 ‖ 2 + ‖ˇx 2 ‖ 2 + ‖ˇx 3 ‖ 2 + ‖ˇx 4 ‖ 2tr(GΦ i1 = ďi1 , i = 2, 3, 4tr ( ) (753)Ge i e T i = ‖ˇxi ‖ 2 , i = 2, 3, 4tr(G(e i e T j + e j e T i )/2) = ˇx T i ˇx j , 2 ≤ i < j = 3, 4which is invertible:⎡svec(I) Tsvec(Φ 21 ) Tsvec(Φ 31 ) Tsvec(Φ 41 ) Tsvec(e 2 e T 2 ) Tsvec G =svec(e 3 e T 3 ) Tsvec(e 4 e T 4 ) Tsvec ( (e⎢ 2 e T 3 + e 3 e T 2 )/2 ) T⎣ svec ( (e 2 e T 4 + e 4 e T 2 )/2 ) Tsvec ( (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 21ď 31ď 41‖ˇx 2 ‖ 2‖ˇx 3 ‖ 2‖ˇx 4 ‖ 2⎢ ˇx T 2ˇx 3⎥⎣ ˇx T 2ˇx 4⎦ˇx T 3ˇx 4(754)That line in the ambient space S 4 of G is traced by the right-hand side.One must show this line to be tangential (2.1.7.2) to S 4 + in order to proveuniqueness. Tangency is possible for affine dimension 1 or 2 while itsoccurrence depends completely on the known measurement data. But as soon as significant noise is introduced or whenever distance data isincomplete, such problems can remain convex although the set of all optimalsolutions generally becomes a convex set bigger than a single point (but stillcontaining the noiseless solution).