v2009.01.01 - Convex Optimization

4.4. RANK-CONSTRAINED SEMIDEFINITE PROGRAM 287
MARKET St.
Figure 75: Uncertainty ellipsoid in R 2 for each of 15 sensors • located within
three city blocks in downtown San Francisco. Data by Polaris Wireless. [279]
problem statement
Ascribe points in a list {x l ∈ R n , l=1... N} to the columns of a matrix X ;
X = [x 1 · · · x N ] ∈ R n×N (68)
where N is regarded as cardinality of list X . Positive semidefinite matrix
X T X , formed from inner product of the list, is a Gram matrix; [215,3.6]
⎡
⎤
‖x 1 ‖ 2 x T 1x 2 x T 1x 3 · · · x T 1x N
x
G = ∆ T 2x 1 ‖x 2 ‖ 2 x T 2x 3 · · · x T 2x N
X T X =
x T 3x 1 x T 3x 2 ‖x 3 ‖ 2 ... x T 3x N
∈ S N + (807)
⎢
⎣
.
. . . .
. ⎥ . . . ⎦
xN Tx 1 xN Tx 2 xN Tx 3 · · · ‖x N ‖ 2
where S N + is the convex cone of N ×N positive semidefinite matrices in the
real symmetric matrix subspace S N .
Existence of noise precludes measured distance from the input data. We
instead assign measured distance to a range estimate specified by individual
upper and lower bounds: d ij is an upper bound on distance-square from i th