v2009.01.01 - Convex Optimization

290 CHAPTER 4. SEMIDEFINITE PROGRAMMING
direction matrix W
Denote by Z ⋆ an optimal composite matrix from semidefinite program (678).
Then for Z ⋆ ∈ S N+n whose eigenvalues λ(Z ⋆ )∈ R N+n are arranged in
nonincreasing order, (Ky Fan)
N+n
∑
λ(Z ⋆ ) i
i=n+1
= minimize
W ∈ S N+n 〈Z ⋆ , W 〉
subject to 0 ≼ W ≼ I
trW = N
(1581a)
which has an optimal solution that is known in closed form. This eigenvalue
sum is zero when Z ⋆ has rank n or less.
Foreknowledge of optimal Z ⋆ , to make possible this search for W , implies
iteration; id est, semidefinite program (678) is solved for Z ⋆ initializing
W = I or W = 0. Once found, Z ⋆ becomes constant in semidefinite
program (1581a) where a new normal direction W is found as its optimal
solution. Then the cycle (678) (1581a) iterates until convergence. When
rankZ ⋆ = n , solution via this convex iteration is optimal for sensor-network
localization problem (672) and its equivalent (677).
numerical solution
In all examples to follow, number of anchors
m = √ N (680)
equals square root of cardinality N of list X . Indices set I identifying all
measurable distances • is ascertained from connectivity matrix (673), (674),
(675), or (676). We solve iteration (678) (1581a) in dimension n = 2 for each
respective example illustrated in Figure 71 through Figure 74.
In presence of negligible noise, true position is reliably localized for every
standardized example; noteworthy in so far as each example represents an
incomplete graph. This means the set of all solutions having lowest rank is
a single point, to within a rigid transformation.
To make the examples interesting and consistent with previous work, we
randomize each range of distance-square that bounds 〈G, (e i −e j )(e i −e j ) T 〉
in (678); id est, for each and every (i,j)∈ I
d ij = d ij (1 + √ 3η χ l
) 2
d ij = d ij (1 − √ 3η χ l+1
) 2 (681)