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


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.


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



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).


  1. 1.
    Aguiar e Oliveira, H., Jr., Ingber, L., Petraglia, A., Petraglia, M.R., Machado, M.A.S.: Nonlinear equation solving. In: Stochastic Global Optimization and Its Applications with Fuzzy Adaptive Simulated Annealing. Intelligent Systems Reference Library, vol. 35, pp. 169–187. Springer, Berlin Heidelberg (2012) ISBN 978-3-642-27478-7. doi: 10.1007/978-3-642-27479-4_10
  2. 2.
    Aiex, R.M., Resende, M.G.C., Ribeiro, C.C.: Probability distribution of solution time in GRASP: an experimental investigation. J. Heuristics 8, 343–373 (2002)CrossRefGoogle Scholar
  3. 3.
    Aiex, R.M., Resende, M.G.C., Ribeiro, C.C.: TTTPLOTS: a perl program to create time-to-target plots. Opt. Lett. 1, 355–366 (2007)CrossRefGoogle Scholar
  4. 4.
    Bean, J.C.: Genetic algorithms and random keys for sequencing and optimization. ORSA J. Comput. 6, 154–160 (1994)CrossRefGoogle Scholar
  5. 5.
    Ericsson, M., Resende, M.G.C., Pardalos, P.M.: A genetic algorithm for the weight setting problem in OSPF routing. J. Comb. Opt. 6, 299–333 (2002)CrossRefGoogle Scholar
  6. 6.
    Floudas, C.A.: Recent advances in global optimization for process synthesis, design, and control: enclosure of all solutions. Comput. Chem. Eng. 23, S963–S973 (1999)CrossRefGoogle Scholar
  7. 7.
    Floudas, C.A., Pardalos, P.M., Adjiman, C., Esposito, W., Gumus, Z., Harding, S., Klepeis, J., Meyer, C., Schweiger, C.: Handbook of Test Problems in Local and Global Optimization. Kluwer Academic Publishers, Dordrecht (1999)CrossRefGoogle Scholar
  8. 8.
    Gonçalves, J.F., Almeida, J.: A hybrid genetic algorithm for assembly line balancing. J. Heuristics 8, 629–642 (2002)CrossRefGoogle Scholar
  9. 9.
    Gonçalves, J.F., Resende, M.G.C.: An evolutionary algorithm for manufacturing cell formation. Comput. Ind. Eng. 47, 247–273 (2004)CrossRefGoogle Scholar
  10. 10.
    Hirsch, M.J., Pardalos, P.M., Resende, M.G.C.: Solving systems of nonlinear equations with continuous GRASP. Nonlinear Anal. Real World Appl. 10, 2000–2006 (2009)CrossRefGoogle Scholar
  11. 11.
    Hirsch, M.J., Pardalos, P.M., Resende, M.G.C.: Speeding up continuous GRASP. Eur. J. Oper. Res. 205, 507–521 (2010)CrossRefGoogle Scholar
  12. 12.
    Lester, I.: Adaptive simulated annealing (asa): lessons learned. Control Cybern. 25, 33–54 (1996)Google Scholar
  13. 13.
    Kearfott, R.B.: Some tests on generalized bisection. ACM Trans. Math. Softw. 13(3), 197–220 (1987)CrossRefGoogle Scholar
  14. 14.
    Kearfott, R.B., Novoa, M.: Algorithm 681: intbis, a portable interval newton/bisection package. ACM Trans. Math. Softw. 16(2):152–157 (1990) doi: 10.1145/78928.78931 Google Scholar
  15. 15.
    Kubicek, M., Hofmann, H., Hlavacek, V., Sinkule, J.: Multiplicity and stability in a sequence of two nonadiabatic nonisothermal CSTR. Chem. Eng. Sci. 35, 987–996 (1980)CrossRefGoogle Scholar
  16. 16.
    Matsumoto, M., Nishimura, T.: Mersenne twister: a 623-dimensionally equidistributed uniform pseudo-random number generator. ACM Trans. Model. Comput. Simul. 8(1), 3–30 (1998)CrossRefGoogle Scholar
  17. 17.
    Merlet, J.P.: The COPRIN examples page. (2006)
  18. 18.
    Oliveira Jr, H.: Fuzzy control of stochastic global optimization algorithms and VSFR. Naval Res. Mag. 16, 103–113 (2003)Google Scholar
  19. 19.
    Pramanik, S.: Kinematic synthesis of a six-member mechanism for automotive steering. ASME J. Mech. Design 124, 642–645 (2002)CrossRefGoogle Scholar
  20. 20.
    R Development Core Team. R: A language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria (2011). URL: ISBN 3-900051-07-0
  21. 21.
    Reklaitis, G., Ragsdell, K.: Engineering Optimization. Wiley, New York (1983)Google Scholar
  22. 22.
    Royden, H.L.: Real Analysis, 3rd edn. Macmillan Publishing, New York (1988)Google Scholar
  23. 23.
    Spears, W.M., DeJong K.A.: On the virtues of parameterized uniform crossover. In: Proceedings of the Fourth International Conference on Genetic Algorithms, pp. 230–236 (1991)Google Scholar
  24. 24.
    Tsai, L.W., Morgan, A.P.: Solving the kinematics of the most general six- and five-degree-of-freedom manipulators by continuation methods. J. Mech. Trans. Autom. Design 107, 189–200 (1985)CrossRefGoogle Scholar

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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