Numerical Optimal Control of Integral-Algebraic Equations Using Differential Evolution with Fletcher’s Filter

  • Wojciech Rafajłowicz
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8467)


Integral-algebraic equations are an interesting method of modeling real world problems with not too severe assumptions. We proposed a simple numerical method of using differential evolution. Constraints in optimal control problems are handled using a method based on the works of Fletcher and his co-workers’ filter.

Numerical results for typical benchmark problems are provided. The efficiency of the proposed method occurred to be satisfactory.


Optimal Control Problem Pareto Front Sequential Quadratic Programming Goal Function Simple Numerical Method 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Brunner, H.: Collocation Methods for Volterra Integral and Related Functional Differential Equations. In: Cambridge Monographs on Applied and Computational Mathematics. Cambridge University Press (2004)Google Scholar
  2. 2.
    Chiou, J.-P., Wang, F.: A hybrid method of differential evolution with application to optimal control problems of a bioprocess system. In: Proceedings of the IEEE World Congress on Computational Intelligence, The 1998 IEEE International Conference on Evolutionary Computation, pp. 627–632 (1998)Google Scholar
  3. 3.
    Cpałka, K., Rutkowski, L.: Evolutionary learning of flexible neuro-fuzzy structures. In: Recent Advances in Control and Automation, pp. 398–407. Akademicka Oficyna Wydawnicza EXIT (2008)Google Scholar
  4. 4.
    Fletcher, R., Leyffer, S., Toint, P.L.: On the global convergence of a filter-SQP algorithm. SIAM J. Optim. 13, 44–59 (2002)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Fletcher, R., Gould, N.I.M., Leyffer, S., Toint, P.L., Wächter, A.: Global convergence of trust-region SQP-filter algorithms for general nonlinear programming. SIAM J. Optimization 13, 635–659 (2002)CrossRefzbMATHGoogle Scholar
  6. 6.
    Fletcher, R.: A Sequential Linear Constraint Programming algorithm for NLP. SIAM Journal of Optimization Vol (3), 772–794Google Scholar
  7. 7.
    Galar, R.: Handicapped Individua in Evolutionary Processes. Biol. Cybern. 53, 1–9 (1985)CrossRefzbMATHGoogle Scholar
  8. 8.
    Galar, R.: Evolutionary Search with Soft Selection. Biol. Cybern. 60, 357–364 (1989)zbMATHMathSciNetGoogle Scholar
  9. 9.
    Gong, W., Cai, Z.: A Multiobjective Differential Evolution Algorithm for Constrained Optimization. In: IEEE Congress on Evolutionary Computation (CEC 2008) (2008)Google Scholar
  10. 10.
    Gordián-Rivera, L.-A., Mezura-Montes, E.: A Combination of Specialized Differential Evolution Variants for Constrained Optimization. In: Pavón, J., Duque-Méndez, N.D., Fuentes-Fernández, R. (eds.) IBERAMIA 2012. LNCS, vol. 7637, pp. 261–270. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  11. 11.
    Kauthen, J.-P.: The numerical solution of integral-algebraic equations of index 1 by polynomial spline collocation methods. Mathematics of Computation 70(236), 1503–1514 (2000)CrossRefMathSciNetGoogle Scholar
  12. 12.
    Kress, R.: Linear Integral Equations. Applied Mathematical Sciences, vol. 82. Springer (1989)Google Scholar
  13. 13.
    Lopez Cruz, I.L., Van Willigenburg, L.G., Van Straten, G.: Efficient Differential Evolution algorithms for multimodal optimal control problems. Applied Soft Computing 3(2), 97–122 (2003) ISSN 1568-4946Google Scholar
  14. 14.
    de Melo, V., Grazieli, L., Costa, C.: Evaluating differential evolution with penalty function to solve constrained engineering problems. Expert Systems with Applications 39, 7860–7863 (2012)CrossRefGoogle Scholar
  15. 15.
    Mezura-Montes, E., Coello, C.A.: A Simple Multimembered Evolution Strategy to Solve Constrained Optimization Problems. IEEE Transactions on Evolutionary Computation 9(1), 1–17 (2005)Google Scholar
  16. 16.
    Mezura-Montes, E., Coello Coello, C.A., Tun-Morales, E.I.: Simple Feasibility Rules and Differential Evolution for Constrained Optimization. In: Monroy, R., Arroyo-Figueroa, G., Sucar, L.E., Sossa, H. (eds.) MICAI 2004. LNCS (LNAI), vol. 2972, pp. 707–716. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  17. 17.
    Storn, R., Price, K.: Differential evolution – a simple and efficient adaptive scheme for global optimization over continuous spaces. Technical report (1995)Google Scholar
  18. 18.
    Storn, R., Price, K.: Differential evolution – a simple and efficient heuristic for global optimization over continuous spaces. Journal of Global Optimization 11, 341–359 (1997)CrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    Price, K., Storn, R., Lampinen, J.: Differential Evolution A Practical Approach to Global Optimization. Springer, Heidelberg (2005)zbMATHGoogle Scholar
  20. 20.
    Rafajłowicz, E., Styczeń, K., Rafajłowicz, W.: A modified filter SQP method as a tool for optimal control of nonlinear systems with spatio-temporal dynamics. International Journal of Applied Mathematics and Computer Science 22(2) (2012)Google Scholar
  21. 21.
    Rafajłowicz, E., Rafajłowicz, W.: Fletcher’s Filter Methodology as a Soft Selector in Evolutionary Algorithms for Constrained Optimization. In: Rutkowski, L., Korytkowski, M., Scherer, R., Tadeusiewicz, R., Zadeh, L.A., Zurada, J.M. (eds.) EC 2012 and SIDE 2012. LNCS, vol. 7269, pp. 333–341. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  22. 22.
    Rafajłowicz, W.: Method of handling constraints in differential evolution using fletcher’s filter. In: Rutkowski, L., Korytkowski, M., Scherer, R., Tadeusiewicz, R., Zadeh, L.A., Zurada, J.M. (eds.) ICAISC 2013, Part II. LNCS (LNAI), vol. 7895, pp. 46–55. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  23. 23.
    Rocha, A.M.A.C., Costa, M.F.P., Fernandes, E.M.G.P.: An Artificial Fish Swarm Filter-Based Method for Constrained Global Optimization. In: Murgante, B., Gervasi, O., Misra, S., Nedjah, N., Rocha, A.M.A.C., Taniar, D., Apduhan, B.O. (eds.) ICCSA 2012, Part III. LNCS, vol. 7335, pp. 57–71. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  24. 24.
    Skowron, M., Styczeń, K.: Evolutionary search for globally optimal constrained stable cycles. Chemical Engineering Science 61(24), 7924–7932 (2006)CrossRefGoogle Scholar
  25. 25.
    Skowron, M., Styczeń, K.: Evolutionary search for globally optimal stable multicycles in complex systems with inventory couplings. International Journal of Chemical Engineering (2009)Google Scholar
  26. 26.
    Wang, F.-S., Chiou, J.-P.: Optimal Control and Optimal Time Location Problems of Differential-Algebraic Systems by Differential Evolution. Ind. Eng. Chem. Res. 36(12), 5348–5357 (1997)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Wojciech Rafajłowicz
    • 1
  1. 1.Wojciech Rafajłowicz is with the Institute of Computer Engineering,Control and RoboticsWrocław University of TechnologyWrocławPoland

Personalised recommendations