v2009.01.01 - Convex Optimization

5.4. EDM DEFINITION 365
By eliminating some equality constraints for this particular problem,
matrix variable dimension can be reduced. From5.8.3 we have
[ ] 0
−VNDV T N = 11 T T 1
− [0 I ] D
(838)
I 2
(which does not hold more generally) where identity matrix I ∈ S N−1 has
one less dimension than EDM D . By defining an EDM principal submatrix
[ ]
ˆD = ∆ 0
T
[0 I ] D ∈ S N−1
I
h
(839)
we get a convex problem equivalent to (834)
minimize − tr(W ˆD)
ˆD∈S N−1
subject to ˆDij ≥ 1 , 1 ≤ i < j = 2... N −1
11 T − ˆD 1 2 ≽ 0
δ( ˆD) = 0
(840)
Any feasible matrix 11 T − ˆD 1 2
belongs to an elliptope (5.9.1.0.1).
This next example shows how finding the common point of intersection
for three circles in a plane, a nonlinear problem, has convex expression.
5.4.2.2.4 Example. So & Ye trilateration in wireless sensor network.
Given three known absolute point positions in R 2 (three anchors ˇx 2 , ˇx 3 , ˇx 4 )
and only one unknown point (one sensor x 1 ∈ R 2 ), the sensor’s absolute
position is determined from its noiseless measured distance-square ď i1
to each of three anchors (Figure 2, Figure 97(a)). This trilateration
can be expressed as a convex optimization problem in terms of list
X = ∆ [x 1 ˇx 2 ˇx 3 ˇx 4 ]∈ R 2×4 and Gram matrix G∈ S 4 (807):
minimize trG
G∈S 4 , X∈R2×4 subject to tr(GΦ i1 ) = ďi1 , i = 2, 3, 4
tr ( )
Ge i e T i = ‖ˇx i ‖ 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
X(:, 2:4) = [ ˇx 2 ˇx 3 ˇx 4 ]
[ ] I X
X T
≽ 0
G
(841)