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Nonlinear Equation Solving

  • Hime Aguiar e Oliveira Junior
  • Lester Ingber
  • Antonio Petraglia
  • Mariane Rembold Petraglia
  • Maria Augusta Soares Machado
Part of the Intelligent Systems Reference Library book series (ISRL, volume 35)

Abstract

This chapter introduces a global optimization approach for finding solutions of nonlinear systems of functional equations using Fuzzy ASA. The original problem is transformed into a global optimization one by synthesizing objective functions whose global minima, if any, are also solutions to the original system. The global minimization process is triggered from different starting points so as to find as many solutions as possible. To demonstrate its utility, the method is applied to several types of equations, presenting very good results. The equation systems are composed of n equations on n-dimensional Euclidean spaces.

Keywords

Global Optimization Merit Function Polynomial System Search Region Solution Versus 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Edgar, G.: Measure, Topology, and Fractal Geometry. Springer, Heidelberg (2008)CrossRefzbMATHGoogle Scholar
  2. 2.
    Effati, S., Nazemi, A.R.: A new method for solving a system of the nonlinear equations. Appl. Math. Comput. 168(2), 877–894 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Grosan, C., Abraham, A.: A new approach for solving nonlinear equations systems. IEEE Transactions on Systems, Man and Cybernetics - Part A: Systems and Humans 38(3), 698–714 (2008)CrossRefGoogle Scholar
  4. 4.
    Grosan, C., Abraham, A.: Multiple solutions for a system of nonlinear equations. International Journal of Innovative Computing, Information and Control 4(9), 2161–2170 (2008)Google Scholar
  5. 5.
    Hirsch, M.J., Pardalos, P.M., Resende, M.G.C.: Solving systems of nonlinear equations with continuous GRASP. Nonlinear Analysis: Real World Applications 10, 2000–2006 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Ingber, L.: Adaptive simulated annealing (ASA): Lessons learned. Control and Cybernetics 25(1), 33–54 (1996)zbMATHGoogle Scholar
  7. 7.
    Moore, R.E.: Methods and Applications of Interval Analysis. SIAM, Philadelphia (1979)CrossRefzbMATHGoogle Scholar
  8. 8.
    Morgan, A.P.: Computing all solutions to polynomial systems using homotopy continuation. Appl. Math. Comput. 24(2), 115–138 (1987)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Morgan, A.P.: Solving Polynomial Systems Using Continuation for Scientific and Engineering Problems. Prentice-Hall, Englewood Cliffs (1987)zbMATHGoogle Scholar
  10. 10.
    Oliveira Jr., H.: Fuzzy control of stochastic global optimization algorithms and VFSR. Naval Research Magazine 16, 103–113 (2003)Google Scholar
  11. 11.
    Pachter, R., Wang, Z.: Adaptive Simulated Annealing and its Application to Protein Folding. In: Floudas, C.A., Pardalos, P.M. (eds.) Encyclopedia of Optimization. Springer, Heidelberg (2009)Google Scholar
  12. 12.
    Verschelde, J., Verlinden, P., Cools, R.: Homotopies exploiting Newton polytopes for solving sparse polynomial systems. SIAM J. Numer. Anal. 31(3), 915–930 (1994)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag GmbH Berlin Heidelberg 2012

Authors and Affiliations

  • Hime Aguiar e Oliveira Junior
    • 1
  • Lester Ingber
    • 2
  • Antonio Petraglia
    • 3
  • Mariane Rembold Petraglia
    • 3
  • Maria Augusta Soares Machado
    • 4
  1. 1.Rio de JaneiroBrazil
  2. 2.Lester Ingber Research AshlandUSA
  3. 3.Faculdades IBMEC Rio de JaneiroBrazil
  4. 4.IBMEC-RJ Rio de JaneiroBrazil

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