v2007.09.13 - Convex Optimization

26 CHAPTER 1. OVERVIEWoriginalreconstructionFigure 6: About five thousand points along the borders constituting theUnited States were used to create an exhaustive matrix of interpoint distancefor each and every pair of points in the ordered set (a list); called aEuclidean distance matrix. From that noiseless distance information, it iseasy to reconstruct the map exactly via the Schoenberg criterion (724).(5.13.1.0.1, confer Figure 90) Map reconstruction is exact (to within a rigidtransformation) given any number of interpoint distances; the greater thenumber of distances, the greater the detail (just as it is for all conventionalmap preparation).exclusively to a positive semidefinite cone boundary. A three-dimensionalpolyhedral analogue for the positive semidefinite cone of 3 ×3 symmetricmatrices is introduced. This analogue is a new tool for visualizing coexistenceof low- and high-rank optimal solutions in 6 dimensions. We find aminimum-cardinality Boolean solution to an instance of Ax = b :minimize ‖x‖ 0xsubject to Ax = bx i ∈ {0, 1} ,i=1... n(576)The sensor-network localization problem is solved in any dimension in thischapter. We introduce the method of convex iteration for constraining rankin the form rankG ≤ ρ for some semidefinite programs in G .