v2009.01.01 - Convex Optimization

5.4. EDM DEFINITION 373
corrected by choosing a different normal for the linear objective function,
now implicitly the identity matrix I ; id est,
trG = 〈G , I 〉 ← 〈G , δ(u)〉 (850)
where vector u ∈ R 5 is randomly selected. A random search for a good
normal δ(u) in only a few iterations is quite easy and effective because
the problem is small, an optimal solution is known a priori to exist in two
dimensions, a good normal direction is not necessarily unique, and (we
speculate) because the feasible affine-subset slices the positive semidefinite
cone thinly in the Euclidean sense. 5.16
We explore ramifications of noise and incomplete data throughout; their
individual effect being to expand the optimal solution set, introducing more
solutions and higher-rank solutions. Hence our focus shifts in4.4 to
discovery of a reliable means for diminishing the optimal solution set by
introduction of a rank constraint.
Now we illustrate how a problem in distance geometry can be solved
without equality constraints representing measured distance; instead, we
have only upper and lower bounds on distances measured:
5.4.2.2.8 Example. Wireless location in a cellular telephone network.
Utilizing measurements of distance, time of flight, angle of arrival, or signal
power, multilateration is the process of localizing (determining absolute
position of) a radio signal source • by inferring geometry relative to multiple
fixed base stations ◦ whose locations are known.
We consider localization of a cellular telephone by distance geometry,
so we assume distance to any particular base station can be inferred from
received signal power. On a large open flat expanse of terrain, signal-power
measurement corresponds well with inverse distance. But it is not uncommon
for measurement of signal power to suffer 20 decibels in loss caused by factors
such as multipath interference (signal reflections), mountainous terrain,
man-made structures, turning one’s head, or rolling the windows up in an
automobile. Consequently, contours of equal signal power are no longer
circular; their geometry is irregular and would more aptly be approximated
5.16 The log det rank-heuristic from7.2.2.4 does not work here because it chooses the
wrong normal. Rank reduction (4.1.1.2) is unsuccessful here because Barvinok’s upper
bound (2.9.3.0.1) on rank of G ⋆ is 4.