On Sensor Network Localization Using SDP Relaxation

A Semidefinite Programming (SDP) relaxation is an effective computational method to solve a Sensor Network Localization problem, which attempts to determine the locations of a group of sensors given the distances between some of them [11]. In this paper, we analyze and determine new sufficient conditions and formulations that guarantee that the SDP relaxation is exact, i.e., gives the correct solution. These conditions can be useful for designing sensor networks and managing connectivities in practice. Our main contribution is twofold: We present the first non-asymptotic bound on the connectivity or radio range requirement of the sensors in order to ensure the network is uniquely localizable. Determining this range is a key component in the design of sensor networks, and we provide a result that leads to a correct localization of each sensor, for any number of sensors. Second, we introduce a new class of graphs that can always be correctly localized by an SDP relaxation. Specifically, we show that adding a simple objective function to the SDP relaxation model will ensure that the solution is correct when applied to a triangulation graph. Since triangulation graphs are very sparse, this is informationally efficient, requiring an almost minimal amount of distance information. We also analyze a number objective functions for the SDP relaxation to solve the localization problem for a general graph.Comment: 20 pages, 4 figures, submitted to the Fields Institute Communications Series on Discrete Geometry and Optimizatio

Universal Rigidity: Towards Accurate and Efficient Localization of Wireless Networks

Abstract—A fundamental problem in wireless ad–hoc and sensor networks is that of determining the positions of nodes. Often, such a problem is complicated by the presence of nodes whose positions cannot be uniquely determined. Most existing work uses the notion of global rigidity from rigidity theory to address the non–uniqueness issue. However, such a notion is not entirely satisfactory, as it has been shown that even if a network localization instance is known to be globally rigid, the problem of determining the node positions is still intractable in general. In this paper, we propose to use the notion of universal rigidity to bridge such disconnect. Although the notion of universal rigidity is more restrictive than that of global rigidity, it captures a large class of networks and is much more relevant to the efficient solvability of the network localization problem. Specifically, we show that both the problem of deciding whether a given network localization instance is universally rigid and the problem of determining the node positions of a universally rigid instance can be solved efficiently using semidefinite programming (SDP). Then, we give various constructions of universally rigid instances. In particular, we show that trilateration graphs are generically universally rigid, thus demonstrating not only the richness of the class of universally rigid instances, but also the fact that trilateration graphs possess much stronger geometric properties than previously known. Finally, we apply our results to design a novel edge sparsification heuristic that can reduce the size of the input network while provably preserving its original localization properties. One of the applications of such heuristic is to speed up existing convex optimization–based localization algorithms. Simulation results show that our speedup approach compares very favorably with existing ones, both in terms of accuracy and computation time

Newsvendor Optimization with Limited Distribution Information ∗

We report preliminary results on stochastic optimization with limited distributional information. Lack of complete distribution calls for stochastically robust models that, after exploiting available limited or partial information, offer risk-shielded solutions, i.e., solutions that are insensitive to all possible distributions of random variables. We focus on the well-known newsvendor problem in this study, where the distribution of the random demand is only specified by its mean and one of the following: its standard deviation or its support. We propose a stochastically robust model for the newsvendor problem. More specifically, our model tries to minimize the regret that is defined as the ratio of the expected cost based on limited information to that based on complete information, called Relative Expected Value of Distribution (REVD). We show how to derive an optimal solution to the REVD model. Numerical examples are provided to compare our model with other similar approaches. The goal is to establish a confidence ratio that the decision from our model is not worse, relatively, too much than the decision based on the true distribution which would be never known exactly in real world applications.

An interior-point path-following algorithm for computing a Leontief economy equilibrium

Linear complementarity problem, Homotopy, Arrow-Debreu-Leontief,