v2009.01.01 - Convex Optimization

372 CHAPTER 5. EUCLIDEAN DISTANCE MATRIX
5.4.2.2.7 Example. Tandem trilateration in wireless sensor network.
Given three known absolute point-positions in R 2 (three anchors ˇx 3 , ˇx 4 , ˇx 5 )
and two unknown sensors x 1 , x 2 ∈ R 2 , the sensors’ absolute positions are
determinable from their noiseless distances-square (as indicated in Figure 99)
assuming the anchors exhibit no rotational or reflective symmetry in their
affine hull (5.5.2). This example differs from Example 5.4.2.2.4 in so far
as trilateration of each sensor is now in terms of one unknown position, the
other sensor. We express this localization as a convex optimization problem
(a semidefinite program,4.1) in terms of list X ∆ = [x 1 x 2 ˇx 3 ˇx 4 ˇx 5 ]∈ R 2×5
and Gram matrix G∈ S 5 (807) via relaxation (842):
minimize trG
G∈S 5 , X∈R2×5 subject to tr(GΦ i1 ) = ďi1 , i = 2, 4, 5
tr(GΦ i2 ) = ďi2 , i = 3, 5
tr ( )
Ge i e T i = ‖ˇx i ‖ 2 , i = 3, 4, 5
tr(G(e i e T j + e j e T i )/2) = ˇx T i ˇx j , 3 ≤ i < j = 4, 5
X(:, 3:5) = [ ˇx 3 ˇx 4 ˇx 5 ]
[ ] I X
X T
≽ 0
G
(849)
where
Φ ij = (e i − e j )(e i − e j ) T ∈ S N + (796)
This problem realization is fragile because of the unknown distances between
sensors and anchors. Yet there is no more information we may include beyond
the 11 independent equality constraints on the Gram matrix (nonredundant
constraints not antithetical) to reduce the feasible set 5.14 . (By virtue of their
dimensioning, the sensors are already constrained to R 2 the affine hull of the
anchors.)
Exhibited in Figure 100 are two mistakes in solution X ⋆ (:,1:2) due
to a rank-3 optimal Gram matrix G ⋆ . The trace objective is a heuristic
minimizing convex envelope of quasiconcave function 5.15 rankG. (2.9.2.6.2,
7.2.2.1) A rank-2 optimal Gram matrix can be found and the errors
5.14 the presumably nonempty convex set of all points G and X satisfying the constraints.
5.15 Projection on that nonconvex subset of all N ×N-dimensional positive semidefinite
matrices, in an affine subset, whose rank does not exceed 2 is a problem considered difficult
to solve. [306,4]