v2010.10.26 - Convex Optimization

31Figure 8: Robotic vehicles in concert can move larger objects, guard aperimeter, perform surveillance, find land mines, or localize a plume of gas,liquid, or radio waves. [138]The Schoenberg criterion (910) for identifying a Euclidean distancematrix is revealed to be a discretized membership relation (dual generalizedinequalities, a new Farkas’-like lemma) between the EDM cone and itsordinary dual: EDM N∗ . A matrix criterion for membership to the dualEDM cone is derived that is simpler than the Schoenberg criterion:D ∗ ∈ EDM N∗ ⇔ δ(D ∗ 1) − D ∗ ≽ 0 (1274)There is a concise equality, relating the convex cone of Euclidean distancematrices to the positive semidefinite cone, apparently overlooked in theliterature; an equality between two large convex Euclidean bodies:EDM N = S N h ∩ ( )S N⊥c − S N +(1268)In chapter 7 Proximity problems we explore methods of solutionto a few fundamental and prevalent Euclidean distance matrix proximityproblems; the problem of finding that distance matrix closest, in some sense,