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COOPERATIVE LOCALIZATION: ON MOTIONINDUCED INITIALIZATION AND JOINT STATE ESTIMATION UNDER COMMUNICATION CONSTRAINTS
"... This thesis would not have been possible without the support of a number of people. First of all, my thanks go to my adviser, Professor Stergios Roumeliotis, for his constant encouragement and guidance, for the long hours of passing along his knowledge and experience, for pushing me beyond my own li ..."
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This thesis would not have been possible without the support of a number of people. First of all, my thanks go to my adviser, Professor Stergios Roumeliotis, for his constant encouragement and guidance, for the long hours of passing along his knowledge and experience, for pushing me beyond my own limitations, and for his seemingly endless supply of interesting research problems. I am also thankful for the time and invaluable advice from
REGULARIZATION METHODS FOR SDP RELAXATIONS IN LARGE SCALE POLYNOMIAL OPTIMIZATION
"... We study how to solve semidefinite programming (SDP) relaxations for large scale polynomial optimization. When interiorpoint methods are used, typically only small or moderately large problems could be solved. This paper studies regularization methods for solving polynomial optimization problems. ..."
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We study how to solve semidefinite programming (SDP) relaxations for large scale polynomial optimization. When interiorpoint methods are used, typically only small or moderately large problems could be solved. This paper studies regularization methods for solving polynomial optimization problems. We describe these methods for semidefinite optimization with block structures, and then apply them to solve large scale polynomial optimization problems. The performance is tested on various numerical examples. By regularization methods, significantly bigger problems could be solved on a regular computer, which is almost impossible by interior point methods.
Noisy Sensor Network Localization using Semidefinite Representations and Facial Reduction
, 2010
"... In this paper we extend a recent algorithm for solving the sensor network localization problem (SNL) to include instances with noisy data. In particular, we continue to exploit the implicit degeneracy in the semidefinite programming (SDP) relaxation of SNL. An essential step involves finding good in ..."
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In this paper we extend a recent algorithm for solving the sensor network localization problem (SNL) to include instances with noisy data. In particular, we continue to exploit the implicit degeneracy in the semidefinite programming (SDP) relaxation of SNL. An essential step involves finding good initial estimates for a noisy Euclidean distance matrix, EDM, completion problem. After finding the EDM completion from the noisy data, we rotate the problem using the original positions of the anchors. This is a preliminary working paper, and is a work in progress. Tests are currently ongoing.
Selecting a Monomial Basis for Sums of Squares Programming over a Quotient Ring
"... Abstract — In this paper we describe a method for choosing a “good ” monomial basis for a sums of squares (SOS) program formulated over a quotient ring. It is known that the monomial basis need only include standard monomials with respect to a Groebner basis. We show that in many cases it is possibl ..."
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Abstract — In this paper we describe a method for choosing a “good ” monomial basis for a sums of squares (SOS) program formulated over a quotient ring. It is known that the monomial basis need only include standard monomials with respect to a Groebner basis. We show that in many cases it is possible to use a reduced subset of standard monomials by combining Groebner basis techniques with the wellknown Newton polytope reduction. This reduced subset of standard monomials yields a smaller semidefinite program for obtaining a certificate of nonnegativity of a polynomial on an algebraic variety. I.
Determining 3D Relative Transformations for Any Combination of Range and Bearing Measurements
"... Abstract—In this paper, we address the problem of motioninduced 3D robottorobot extrinsic calibration based on egomotion estimates and combinations of interrobot measurements (i.e., distance and/or bearing observations from either or both of the two robots, recorded across multiple time steps). ..."
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Abstract—In this paper, we address the problem of motioninduced 3D robottorobot extrinsic calibration based on egomotion estimates and combinations of interrobot measurements (i.e., distance and/or bearing observations from either or both of the two robots, recorded across multiple time steps). In particular, we focus on solving minimal problems where the unknown 6degreesoffreedom (DOF) transformation between the two robots is determined based on the minimum number of measurements necessary for finding a finite set of solutions. In order to address the very large number of possible combinations of interrobot observations, we identify symmetries in the measurement sequence and use them to prove that any extrinsic robottorobot calibration problem can be solved based on the solutions of only 14 (base) minimal problems. Moreover, we provide algebraic (closedform) and efficient symbolicnumerical (analytical) solution methods to these minimal problems. Finally, we evaluate the performance of our proposed solvers through extensive simulations and experiments. I.
Comparing SOS and SDP relaxations of sensor network localization
, 2010
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COLLABORATIVE
"... Abstract—In this work, we propose a settheoretic approach to collaborative position location for wireless networks. The proposed method borrows the concept from the parallel projection method (PPM), originally developed for signal recovery with inconsistent convex feasibility sets, modifies and ext ..."
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Abstract—In this work, we propose a settheoretic approach to collaborative position location for wireless networks. The proposed method borrows the concept from the parallel projection method (PPM), originally developed for signal recovery with inconsistent convex feasibility sets, modifies and extends the technique to an iterative and distributed numerical algorithm to estimate node locations, based on incomplete and noisy internode distance estimates. We demonstrate that in the case of noncollaborative position location, the proposed method is analytically equivalent to the parallel implementation of Kaczmarz Algorithm that is guaranteed to converge to a local minimizer and thus a stationary point. For collaborative position location, the proposed iterative PPM is computationally much more efficient than existing methods such as SDP and MDSMAP, while achieving comparable or better localization accuracy and robustness to nonlineofsight (NLOS) bias. Finally, our proposed method can be implemented in a parallel and distributed fashion, and is scalable for large network deployment. Index Terms—Collaborative position location, parallel projection method, Kaczmarz Algorithm, nonlineofsight. Ç
A Potential Reduction Method for Canonical Duality, with an Application to the Sensor Network Localization Problem. Submitted to J. of Global Optim. Preprint available at http://arxiv.org/pdf/1403.5991.pdf
, 2014
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Sensor Network Localization Problems
, 2009
"... Abstract. SFSDP is a Matlab package for solving a sensor network localization problem. These types of problems arise in monitoring and controlling applications using wireless sensor networks. SFSDP implements the semidefinite programming (SDP) relaxation proposed in Kim et al. [2009] for sensor netw ..."
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Abstract. SFSDP is a Matlab package for solving a sensor network localization problem. These types of problems arise in monitoring and controlling applications using wireless sensor networks. SFSDP implements the semidefinite programming (SDP) relaxation proposed in Kim et al. [2009] for sensor network localization problems, as a sparse version of the full semidefinite programming relaxation (FSDP) by Biswas and Ye [2004]. To improve the efficiency of FSDP, SFSDP exploits the aggregated and correlative sparsity of a sensor network localization problem. As a result, SFSDP can handle much largersized problems than other softwares, and threedimensional anchorfree problems. SFSDP can analyze the input data of a sensor network localization problem, solves the problem, and displays the computed locations of sensors. SFSDP also includes the features of generating test problems for numerical experiments. Key words. Sensor network localization problems, semidefinite programming relaxation, sparsity exploitation, Matlab software package.
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"... The notion of convexity underlies important results in many parts of mathematics such as optimization, analysis, combinatorics, probability and number theory. The geometric foundations of the theory of convex sets date back to work of Minkowski, Carathéodory, and Fenchel around 1900. Since then, thi ..."
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The notion of convexity underlies important results in many parts of mathematics such as optimization, analysis, combinatorics, probability and number theory. The geometric foundations of the theory of convex sets date back to work of Minkowski, Carathéodory, and Fenchel around 1900. Since then, this area has expanded into a large number of directions and now includes topics such as highdimensional spaces, convex analysis, polyhedral geometry, computational convexity, approximation methods and others. In the context of optimization, both theory and empirical evidence show that problems with convex constraints allow efficient algorithms. Many applications in the sciences and engineering involve optimization, and it is always extremely advantageous when the underlying feasible regions are convex and have practically useful representations as convex sets. A situation in which convexity has been wellunderstood is the study of convex polyhedra, which are the solution sets of finitely many linear inequalities [27, 86]. A context in algebraic geometry in which convexity arises is the theory of toric varieties. These are algebraic varieties derived from polyhedra [49, 73]. Both convex polyhedra and toric varieties have satisfactory computational techniques associated to them. Linear optimization over polyhedra is linear programming which admits interiorpoint algorithms that run in polynomial time. More generally, polyhedra can be