v2009.01.01 - Convex Optimization

5.4. EDM DEFINITION 367
where
Φ ij = (e i − e j )(e i − e j ) T ∈ S N + (796)
and where the constraint on distance-square ďi1 is equivalently written as
a constraint on the Gram matrix via (809). There are 9 independent affine
equality constraints on that Gram matrix while the sensor is constrained
only by dimensioning to lie in R 2 . Although trG the objective of
minimization 5.12 insures a solution on the boundary of positive semidefinite
cone S 4 + , for this problem, we claim that the set of feasible Gram matrices
forms a line (2.5.1.1) in isomorphic R 10 tangent (2.1.7.2) to the positive
semidefinite cone boundary. (5.4.2.2.5, confer4.2.1.3)
By Schur complement (A.4,2.9.1.0.1), any feasible G and X provide
G ≽ X T X (842)
which is a convex relaxation of the desired (nonconvex) equality constraint
[ ] [ ] I X I [I X ]
X T =
G X T (843)
expected positive semidefinite rank-2 under noiseless conditions.
by (1391), the relaxation admits
But,
(3 ≥) rankG ≥ rankX (844)
(a third dimension corresponding to an intersection of three spheres,
not circles, were there noise). If rank of an optimal solution equals 2,
[ ]
I X
rank
⋆
X ⋆T G ⋆ = 2 (845)
then G ⋆ = X ⋆T X ⋆ by Theorem A.4.0.0.4.
As posed, this localization problem does not require affinely independent
(Figure 23, three noncollinear) anchors. Assuming the anchors exhibit
no rotational or reflective symmetry in their affine hull (5.5.2) and
assuming the sensor x 1 lies in that affine hull, then sensor position solution
x ⋆ 1= X ⋆ (:, 1) is unique under noiseless measurement. [278]
5.12 Trace (trG = 〈I , G〉) minimization is a heuristic for rank minimization. (7.2.2.1)
It may be interpreted as squashing G which is bounded below by X T X as in (842);
id est, G−X T X ≽ 0 ⇒ trG ≥ tr X T X (1389). δ(G−X T X)=0 ⇔ G=X T X (A.7.2)
⇒ tr G = tr X T X which is a condition necessary for equality.