v2010.10.26 - Convex Optimization

5.4. EDM DEFINITION 425now implicitly the identity matrix I ; id est,trG = 〈G , I 〉 ← 〈G , δ(u)〉 (945)where vector u ∈ R 5 is randomly selected. A random search for a goodnormal 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 twodimensions, a good normal direction is not necessarily unique, and (wespeculate) because the feasible affine-subset slices the positive semidefinitecone thinly in the Euclidean sense. 5.19We explore ramifications of noise and incomplete data throughout; theirindividual effect being to expand the optimal solution set, introducing moresolutions and higher-rank solutions. Hence our focus shifts in4.4 todiscovery of a reliable means for diminishing the optimal solution set byintroduction of a rank constraint.Now we illustrate how a problem in distance geometry can be solvedwithout equality constraints representing measured distance; instead, wehave only upper and lower bounds on distances measured:5.4.2.3.9 Example. Wireless location in a cellular telephone network.Utilizing measurements of distance, time of flight, angle of arrival, or signalpower in the context of wireless telephony, multilateration is the processof localizing (determining absolute position of) a radio signal source • byinferring geometry relative to multiple fixed base stations ◦ whose locationsare known.We consider localization of a cellular telephone by distance geometry,so we assume distance to any particular base station can be inferred fromreceived signal power. On a large open flat expanse of terrain, signal-powermeasurement corresponds well with inverse distance. But it is not uncommonfor measurement of signal power to suffer 20 decibels in loss caused by factorssuch as multipath interference (signal reflections), mountainous terrain,man-made structures, turning one’s head, or rolling the windows up in anautomobile. Consequently, contours of equal signal power are no longercircular; their geometry is irregular and would more aptly be approximated5.19 The log det rank-heuristic from7.2.2.4 does not work here because it chooses thewrong normal. Rank reduction (4.1.2.1) is unsuccessful here because Barvinok’s upperbound (2.9.3.0.1) on rank of G ⋆ is 4.