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Canonical Dual Solutions to Sum of Fourth-Order Polynomials Minimization Problems with Applications to Sensor Network Localization

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Sensors: Theory, Algorithms, and Applications

Part of the book series: Springer Optimization and Its Applications ((SOIA,volume 61))

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Abstract

This chapter presents a canonical dual approach for solving a general sum of fourth-order polynomial minimization problem. This problem arises extensively in engineering and science, including database analysis, computational biology, sensor network communications, nonconvex mechanics, and ecology. We first show that this global optimization problem is actually equivalent to a discretized minimal potential variational problem in large deformation mechanics. Therefore, a general analytical solution is proposed by using the canonical duality theory developed by the first author. Both global and local extremality properties of this analytical solution are identified by a triality theory. Application to sensor network localization problem is illustrated. Our results show when the problem is not uniquely localizable, the “optimal solution” obtained by the SDP method is actually a local maximizer of the total potential energy. However, by using a perturbed canonical dual approach, a class of Euclidean distance problems can be converted to a unified concave maximization dual problem with zero duality gap, which can be solved by well-developed convex minimization methods. This chapter should bridge an existing gap between nonconvex mechanics and global optimization.

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Acknowledgements

The work of David Gao and N. Ruan was supported by NSF grant CCF-0514768, US Air Force (AFOSR) grants FA9550-09-1-0285 and FA9550-10-1-0487. The first author is grateful to Drs. Jiapu Zhang and Changzhi Wu at the University of Ballarat for the helpful comments and suggestions.

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Authors and Affiliations

  1. School of Science, Information Technology, and Engineering, University of Ballarat, Mt. Helen, VIC, 3350, Australia

    David Yang Gao

  2. Curtin University of Technology, Perth, WA, 6845, Australia

    Ning Ruan

  3. Department of Industrial and System Engineering, University of Florida, Gainesville, FL, 32611, USA

    Panos M. Pardalos

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  1. David Yang Gao
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  2. Ning Ruan
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  3. Panos M. Pardalos
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Correspondence to David Yang Gao .

Editor information

Editors and Affiliations

  1. Department of Industrial and Systems Eng, University of Florida, Weil Hall 303, Gainesville, 32611, Florida, 32611, USA

    Vladimir L. L. Boginski

  2. Munitions Directorate, Air Force Research Laboratory, Eglin Air Force Base, West Eglin Blvd 101, Eglin AFB, 32542, Florida, 32542, USA

    Clayton W. W. Commander

  3. Dept. Industrial & Systems, Engineering, University of Florida, Weil Hall 303, Gainesville, 32611-6595, Florida, 94305, USA

    Panos M. M. Pardalos

  4. School of Engineering, Department of Management Science and Eng, Stanford University, Via Ortega 475, Stanford, 94305, California, USA

    Yinyu Ye

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Gao, D.Y., Ruan, N., Pardalos, P.M. (2012). Canonical Dual Solutions to Sum of Fourth-Order Polynomials Minimization Problems with Applications to Sensor Network Localization. In: Boginski, V.L., Commander, C.W., Pardalos, P.M., Ye, Y. (eds) Sensors: Theory, Algorithms, and Applications. Springer Optimization and Its Applications(), vol 61. Springer, New York, NY. https://doi.org/10.1007/978-0-387-88619-0_3

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