v2006.03.09 - Convex Optimization

40 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 (479).(4.13.1.0.1, confer Figure 70) 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).Chapter 3, Geometry of convex functions, observes analogiesbetween convex sets and functions: The set of all vector-valued convexfunctions of particular dimension is a closed convex cone. Included amongthe examples in this chapter, we show how the real affine function relatesto convex functions as the hyperplane relates to convex sets. Here, also,pertinent results for multidimensional convex functions are presentedthat are largely ignored in the literature; tricks and tips for determiningtheir convexity and discerning their geometry, particularly with regard tomatrix calculus which remains largely unsystematized when compared withtraditional (ordinary) calculus. Consequently, we collect some results ofmatrix differentiation in the appendices.