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Journal of Global Optimization

, Volume 60, Issue 2, pp 289–306 | Cite as

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

  • Ricardo M. A. Silva
  • Mauricio G. C. Resende
  • Panos M. Pardalos
Article

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.

Keywords

Nonlinear systems of equations Global optimization  Continuous optimization Heuristic Stochastic algorithm Nonlinear programming BRKGA 

Notes

Acknowledgments

The research of R.M.A Silva was partially supported by the Brazilian National Council for Scientific and Technological Development (CNPq), the Foundation for Support of Research of the State of Minas Gerais, Brazil (FAPEMIG), Coordination for the Improvement of Higher Education Personnel, Brazil (CAPES), Foundation for the Support of Development of the Federal University of Pernambuco, Brazil (FADE), the Office for Research and Graduate Studies of the Federal University of Pernambuco (PROPESQ), and the Foundation for Support of Science and Technology of the State of Pernambuco (FACEPE).

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Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Ricardo M. A. Silva
    • 1
  • Mauricio G. C. Resende
    • 2
  • Panos M. Pardalos
    • 3
  1. 1.Centro de InformáticaUniversidade Federal de PernambucoRecife-PEBrazil
  2. 2.Algorithms and Optimization Research DepartmentAT&T Labs ResearchFlorham ParkUSA
  3. 3.Department of Industrial and Systems EngineeringUniversity of FloridaGainesvilleUSA

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