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Global solutions to nonconvex optimization of 4th-order polynomial and log-sum-exp functions

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  • Yi Chen
  • David Gao

Abstract

This paper presents a canonical dual approach for solving a nonconvex global optimization problem governed by a sum of 4th-order polynomial and a log-sum-exp function. Such a problem arises extensively in engineering and sciences. Based on the canonical duality–triality theory, this nonconvex problem is transformed to an equivalent dual problem, which can be solved easily under certain conditions. We proved that both global minimizer and the biggest local extrema of the primal problem can be obtained analytically from the canonical dual solutions. As two special cases, a quartic polynomial minimization and a minimax problem are discussed. Existence conditions are derived, which can be used to classify easy and relative hard instances. Applications are illustrated by several nonconvex and nonsmooth examples. Copyright Springer Science+Business Media New York 2016

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  • Yi Chen & David Gao, 2016. "Global solutions to nonconvex optimization of 4th-order polynomial and log-sum-exp functions," Journal of Global Optimization, Springer, vol. 64(3), pages 417-431, March.
    • Handle: RePEc:spr:jglopt:v:64:y:2016:i:3:p:417-431
    DOI: 10.1007/s10898-014-0244-5
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    References listed on IDEAS

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    1. E. Y. Pee & J. O. Royset, 2011. "On Solving Large-Scale Finite Minimax Problems Using Exponential Smoothing," Journal of Optimization Theory and Applications, Springer, vol. 148(2), pages 390-421, February.
    2. David Yang Gao & Ning Ruan & Hanif D. Sherali, 2010. "Canonical Dual Solutions for Fixed Cost Quadratic Programs," Springer Optimization and Its Applications, in: Altannar Chinchuluun & Panos M. Pardalos & Rentsen Enkhbat & Ider Tseveendorj (ed.), Optimization and Optimal Control, pages 139-156, Springer.
    3. David Yang Gao & Ning Ruan & Panos M. Pardalos, 2012. "Canonical Dual Solutions to Sum of Fourth-Order Polynomials Minimization Problems with Applications to Sensor Network Localization," Springer Optimization and Its Applications, in: Vladimir L. L. Boginski & Clayton W. W. Commander & Panos M. M. Pardalos & Yinyu Ye (ed.), Sensors: Theory, Algorithms, and Applications, pages 37-54, Springer.
    4. E. Polak & J. O. Royset & R. S. Womersley, 2003. "Algorithms with Adaptive Smoothing for Finite Minimax Problems," Journal of Optimization Theory and Applications, Springer, vol. 119(3), pages 459-484, December.
    5. David Gao & Ning Ruan, 2010. "Solutions to quadratic minimization problems with box and integer constraints," Journal of Global Optimization, Springer, vol. 47(3), pages 463-484, July.
    6. Zhenbo Wang & Shu-Cherng Fang & David Gao & Wenxun Xing, 2012. "Canonical dual approach to solving the maximum cut problem," Journal of Global Optimization, Springer, vol. 54(2), pages 341-351, October.
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    Cited by:

    1. Jin, Zhong & Y. Gao, David, 2017. "On modeling and global solutions for d.c. optimization problems by canonical duality theory," Applied Mathematics and Computation, Elsevier, vol. 296(C), pages 168-181.

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