v2007.09.13 - Convex Optimization

258 CHAPTER 4. SEMIDEFINITE PROGRAMMINGit can never be exceeded by subsequent iterations because existence offeasible G and W having that vector inner-product φ has been establishedsimultaneously in each problem. Because the infimum of vector inner-productof two positive semidefinite matrix variables is zero, the nonincreasingsequence of iterations is thus bounded below hence convergent becauseany bounded monotonic sequence in R is convergent. [188,1.2] [30,1.1]Local convergence to some φ is thereby established.When a rank-n feasible solution to (632) exists, it remains pending toshow under what conditions 〈G ⋆ , W ⋆ 〉=0 (633) is achieved by iterativesolution of semidefinite programs (632) and (1475a). Then pair (G ⋆ , W ⋆ )becomes a fixed-point of iteration.A nonexistent feasible rank-n solution would mean failure to converge bydefinition (633) but, as proved, the convex iteration always converges locallyif not globally. Now, an application:4.4.1.1.2 Example. Sensor-Network Localization and Wireless Location.Heuristic solution proposed by Carter & Jin to a sensor-network localizationproblem appeared in a reputable journal [51] 4.22 despite the heavy relianceon heuristics, limitation to two Euclidean dimensions, and misapplication ofsemidefinite programming (SDP). A large network is partitioned into smallersubnetworks (as small as one sensor) and then semidefinite programming andheuristics called spaseloc are applied to localize each and every partitionby two-dimensional distance geometry. Their partitioning procedure isone-pass, yet termed iterative; a term applicable only in so far as adjoiningpartitions can share localized sensors and anchors (absolute sensor positionsknown a priori). But there is no iteration on the entire network, hencethe term “iterative” is misapplied. As partitions are selected based on“rule sets” (heuristics, not geographics), they also term the partitioningadaptive. But there is no adaptation once a partition is determined; hence,another misapplication of an exacting technical term.One can reasonably argue that semidefinite programming methods areunnecessary for localization of large sensor networks. In the past, thesenonlinear localization problems were solved algebraically and computed by4.22 Despite the fact that his name appears as fourth author, Ye had no involvement inwriting this cited paper nor did he contribute to its content. The paper constitutes Jin’sdissertation for University of Toronto although her name appears as second author.