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Friday, March 6, 2009 |
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A Unified Theorem on Semidefinite Programming Rank Reduction and its Applications |
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We present a unified theorem on semidefinite programming solution rank reduction that provides a unified treatment of and generalizes several well--known results in the literature. In particular, it contains as special cases the Johnson--Lindenstrauss lemma on dimensionality reduction, results on low--distortion embedding into low--dimensional Euclidean space, and approximation results on certain quadratic optimization problems. We also illustrate its applications on semidefinite programming (SDP) based model and method for the position estimation problem in Euclidean distance geometry such as graph realization and wireless sensor network localization. We develop an SDP relaxation model and use the duality theory to derive necessary and/or sufficient conditions for whether a network is "localizable" or not, when the distance measures are accurate. |