v2010.10.26 - Convex Optimization

4.4. RANK-CONSTRAINED SEMIDEFINITE PROGRAM 315In summary, initialize Y then iterate numerical solution of (convex)semidefinite programminimize 〈Z , Y 〉A∈S 3 , B∈S 3 , W ∈R 3×2 , H∈R 2×3 ⎡ ⎤I W T Hsubject to Z = ⎣ W A X ⎦≽ 0H T X T BW ≥ 0H ≥ 0(744)with Y = U ⋆ U ⋆T until convergence (which is global and occurs in very fewiterations for this instance).Now, an application to optimal regulation of affine dimension:4.4.1.2.4 Example. Sensor-Network Localization or Wireless Location.Heuristic solution to a sensor-network localization problem, proposedby Carter, Jin, Saunders, & Ye in [72], 4.26 is limited to two Euclideandimensions and applies semidefinite programming (SDP) to littlesubproblems. There, a large network is partitioned into smaller subnetworks(as small as one sensor − a mobile point, whereabouts unknown) and thensemidefinite programming and heuristics called spaseloc are applied tolocalize each and every partition by two-dimensional distance geometry.Their partitioning procedure is one-pass, yet termed iterative; a termapplicable only in so far as adjoining partitions can share localized sensorsand anchors (absolute sensor positions known a priori). But there isno iteration on the entire network, hence the term “iterative” is perhapsinappropriate. As partitions are selected based on “rule sets” (heuristics, notgeographics), they also term the partitioning adaptive. But no adaptation ofa partition actually occurs once it has been determined.One can reasonably argue that semidefinite programming methods areunnecessary for localization of small partitions of large sensor networks.[278] [86] In the past, these nonlinear localization problems were solvedalgebraically and computed by least squares solution to hyperbolic equations;4.26 The paper constitutes Jin’s dissertation for University of Toronto [216] although hername appears as second author. Ye’s authorship is honorary.