v2009.01.01 - Convex Optimization

288 CHAPTER 4. SEMIDEFINITE PROGRAMMING
to j th sensor, while d ij is a lower bound. These bounds become the input
data. Each measurement range is presumed different from the others because
of measurement uncertainty; e.g., Figure 75.
Our mathematical treatment of anchors and sensors is not
dichotomized. 4.28 A known sensor position to high accuracy ˇx i is an
anchor. Then the sensor-network localization problem (672) can be
expressed equivalently: Given a number of anchors m , and I a set of
indices (corresponding to all measurable distances •), for 0 < n < N
minimize trZ
G∈S N , X∈R n×N
subject to d ij ≤ 〈G , (e i − e j )(e i − e j ) T 〉 ≤ d ij ∀(i,j)∈ I
〈G , e i e T i 〉 = ‖ˇx i ‖ 2 , i = N − m + 1... N
〈G , (e i e T j + e j e T i )/2〉 = ˇx T i ˇx j , i < j , ∀i,j∈{N − m + 1... N}
X(:, N − m + 1:N) = [ ˇx N−m+1 · · · ˇx N ]
[ ] I X
Z =
X T
≽ 0
G
rankZ
= n
where e i is the i th member of the standard basis for R N . Distance-square
(677)
d ij = ‖x i − x j ‖ 2 2
∆
= 〈x i − x j , x i − x j 〉 (794)
is related to Gram matrix entries G ∆ =[g ij ] by vector inner-product
d ij = g ii + g jj − 2g ij
= 〈G , (e i − e j )(e i − e j ) T 〉 ∆ = tr(G T (e i − e j )(e i − e j ) T )
(809)
hence the scalar inequalities. The objective function trZ is a heuristic whose
sole purpose is to represent the convex envelope of rankZ . (7.2.2.1.1) By
Schur complement (A.4), any feasible G and X provide a comparison with
respect to the positive semidefinite cone
G ≽ X T X (842)
4.28 Wireless location problem thus stated identically; difference being: fewer sensors.