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Finding multiple roots of a box-constrained system of nonlinear equations with a biased random-key genetic algorithm

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  • Ricardo Silva
  • Mauricio Resende
  • Panos Pardalos

Abstract

Several numerical methods for solving nonlinear systems of equations assume that derivative information is available. Furthermore, these approaches usually do not consider the problem of finding all solutions to a nonlinear system. Rather, most methods output a single solution. In this paper, we address the problem of finding all roots of a system of equations. Our method makes use of a biased random-key genetic algorithm (BRKGA). Given a nonlinear system, we construct a corresponding optimization problem, which we solve multiple times, making use of a BRKGA, with areas of repulsion around roots that have already been found. The heuristic makes no use of derivative information. We illustrate the approach on seven nonlinear equations systems with multiple roots from the literature. Copyright Springer Science+Business Media New York 2014

Suggested Citation

  • Ricardo Silva & Mauricio Resende & Panos Pardalos, 2014. "Finding multiple roots of a box-constrained system of nonlinear equations with a biased random-key genetic algorithm," Journal of Global Optimization, Springer, vol. 60(2), pages 289-306, October.
    • Handle: RePEc:spr:jglopt:v:60:y:2014:i:2:p:289-306
    DOI: 10.1007/s10898-013-0105-7
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    References listed on IDEAS

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    1. Hirsch, M.J. & Pardalos, P.M. & Resende, M.G.C., 2010. "Speeding up continuous GRASP," European Journal of Operational Research, Elsevier, vol. 205(3), pages 507-521, September.
    2. M. Ericsson & M.G.C. Resende & P.M. Pardalos, 2002. "A Genetic Algorithm for the Weight Setting Problem in OSPF Routing," Journal of Combinatorial Optimization, Springer, vol. 6(3), pages 299-333, September.
    3. L. Ingber, 2012. "Adaptive simulated annealing," Lester Ingber Papers 12as, Lester Ingber.
    4. H.A. Oliveira, Jr. & A. Petraglia & L. Ingber & M.A.S. Machado & M.R. Petraglia, . "Stochastic global optimization and its applications with fuzzy adaptive simulated annealing," Lester Ingber Books, Lester Ingber, number 12a2, December-.
    5. James C. Bean, 1994. "Genetic Algorithms and Random Keys for Sequencing and Optimization," INFORMS Journal on Computing, INFORMS, vol. 6(2), pages 154-160, May.
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    Cited by:

    1. Alla Kammerdiner & Alexander Semenov & Eduardo L. Pasiliao, 2022. "Multidimensional Assignment Problem for Multipartite Entity Resolution," Journal of Global Optimization, Springer, vol. 84(2), pages 491-523, October.
    2. Londe, Mariana A. & Pessoa, Luciana S. & Andrade, Carlos E. & Resende, Mauricio G.C., 2025. "Biased random-key genetic algorithms: A review," European Journal of Operational Research, Elsevier, vol. 321(1), pages 1-22.
    3. Jun Pei & Zorica Dražić & Milan Dražić & Nenad Mladenović & Panos M. Pardalos, 2019. "Continuous Variable Neighborhood Search (C-VNS) for Solving Systems of Nonlinear Equations," INFORMS Journal on Computing, INFORMS, vol. 31(2), pages 235-250, April.
    4. Gisela C V Ramadas & Ana Maria A C Rocha & Edite M G P Fernandes, 2015. "Testing Nelder-Mead Based Repulsion Algorithms for Multiple Roots of Nonlinear Systems via a Two-Level Factorial Design of Experiments," PLOS ONE, Public Library of Science, vol. 10(4), pages 1-30, April.

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