v2009.01.01 - Convex Optimization

4.4. RANK-CONSTRAINED SEMIDEFINITE PROGRAM 283
2
3
1
4
Figure 71: 2-lattice in R 2 , hand-drawn. Nodes 3 and 4 are anchors;
remaining nodes are sensors. Radio range of sensor 1 indicated by arc.
0 • ? •
• 0 • •
? • 0 ◦
• • ◦ 0
(673)
Matrix entries dot • indicate measurable distance between nodes while
unknown distance is denoted by ? (question mark). Matrix entries
hollow dot ◦ represent known distance between anchors (to very high
accuracy) while zero distance is denoted 0. Because measured distances
are quite unreliable in practice, our solution to the localization problem
substitutes a distinct range of possible distance for each measurable distance;
equality constraints exist only for anchors.
Anchors are chosen so as to increase difficulty for algorithms dependent
on existence of sensors in their convex hull. The challenge is to find a solution
in two dimensions close to the true sensor positions given incomplete noisy
intersensor distance information.