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A Hybrid Algorithm for the Simple Cell Mapping Method in Multi-objective Optimization

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EVOLVE - A Bridge between Probability, Set Oriented Numerics, and Evolutionary Computation IV

Part of the book series: Advances in Intelligent Systems and Computing ((AISC,volume 227))

Abstract

This paper presents a hybrid gradient free-gradient (GFG) algorithm for the simple cell mapping (SCM) method for multi-objective optimization problems (MOPs). The SCM method is briefly reviewed in the context of the multi-objective optimization problems (MOPs). We present a mixed application of gradient free directed search and gradient search algorithms for the SCM method and discuss its potentials for higher dimensional MOPs. We present several numerical exmaples to demonstrate the effectiveness of the proposed hybrid algorithm. The examples include two simple geometric MOPs, an example with five design parameters, and a proportional-integral-derivative (PID) control design for a second order linear system.

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References

  1. Lara, A., Alvarado, S., Salomon, S., Avigad, G., Coello Coello, C.A., Schütze, O.: The gradient free directed search method as local search within multi-objective evolutionary algorithms. In: Schütze, O., Coello Coello, C.A., Tantar, A.-A., Tantar, E., Bouvry, P., Del Moral, P., Legrand, P. (eds.) EVOLVE - A Bridge Between Probability, Set Oriented Numerics, and Evolutionary Computation II. AISC, vol. 175, pp. 153–168. Springer, Heidelberg (2012)

    Chapter  Google Scholar 

  2. Coello Coello, C.A., Lamont, G., Van Veldhuizen, D.A.: Evolutionary Algorithms for Solving Multi-Objective Problems. Springer, New York (2007)

    MATH  Google Scholar 

  3. Schütze, O., Vasile, M., Coello Coello, C.A.: Computing the set of epsilon-efficient solutions in multiobjective space mission design. Journal of Aerospace Computing, Information, and Communication 8(3), 53–70 (2011)

    Article  Google Scholar 

  4. Dellnitz, M., Schütze, O., Hestermeyer, T.: Covering Pareto sets by multilevel subdivision techniques. Journal of Optimization Theory and Applications 124, 113–155 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  5. Jahn, J.: Multiobjective search algorithm with subdivision technique. Computational Optimization and Applications 35(2), 161–175 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  6. Schütze, O., Vasile, M., Junge, O., Dellnitz, M., Izzo, D.: Designing optimal low thrust gravity assist trajectories using space pruning and a multi-objective approach. Engineering Optimization 41(2), 155–181 (2009)

    Article  MathSciNet  Google Scholar 

  7. Hsu, C.S.: Cell-to-Cell Mapping, A Method of Global Analysis for Nonlinear Systems. Springer, New York (1987)

    Book  MATH  Google Scholar 

  8. Hsu, C.S.: A theory of cell-to-cell mapping dynamical systems. Journal of Applied Mechanics 47, 931–939 (1980)

    Article  MATH  Google Scholar 

  9. Hsu, C.S.: A discrete method of optimal control based upon the cell state space concept. Journal of Optimization Theory and Applications 46(4), 547–569 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  10. Bursal, F.H., Hsu, C.S.: Application of a cell-mapping method to optimal control problems. International Journal of Control 49(5), 1505–1522 (1989)

    MathSciNet  MATH  Google Scholar 

  11. Crespo, L.G., Sun, J.Q.: Stochastic optimal control of nonlinear dynamic systems via bellman’s principle and cell mapping. Automatica 39(12), 2109–2114 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  12. Flashner, H., Burns, T.F.: Spacecraft momentum unloading: the cell mapping approach. Journal of Guidance, Control and Dynamics 13, 89–98 (1990)

    Article  Google Scholar 

  13. Zhu, W.H., Leu, M.C.: Planning optimal robot trajectories by cell mapping. In: Proceedings of Conference on Robotics and Automation, pp. 1730–1735 (1990)

    Google Scholar 

  14. Wang, F.Y., Lever, P.J.A.: A cell mapping method for general optimum trajectory planning of multiple robotic arms. Robotics and Autonomous Systems 12, 15–27 (1994)

    Article  Google Scholar 

  15. Yen, J.Y.: Computer disk file track accessing controller design based upon cell to cell mapping. In: Proceedings of the American Control Conference (1992)

    Google Scholar 

  16. Crespo, L.G., Sun, J.Q.: Solution of fixed final state optimal control problems via simple cell mapping. Nonlinear Dynamics 23, 391–403 (2000)

    Article  MATH  Google Scholar 

  17. Crespo, L.G., Sun, J.Q.: Optimal control of target tracking via simple cell mapping. Journal of Guidance and Control 24, 1029–1031 (2000)

    Article  Google Scholar 

  18. Crespo, L.G., Sun, J.Q.: Fixed final time optimal control via simple cell mapping. Nonlinear Dynamics 31, 119–131 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  19. Pareto, V.: Manual of Political Economy. The MacMillan Press, London (1971); original edition in French (1927)

    Google Scholar 

  20. Hillermeier, C.: Nonlinear Multiobjective Optimization - A Generalized Homotopy Approach. Birkhäuser, Berlin (2001)

    Book  MATH  Google Scholar 

  21. Fliege, J., Svaiter, B.F.: Steepest descent methods for multicriteria optimization. Mathematical Methods of Operations Research 51(3), 479–494 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  22. Liu, G.P., Yang, J.B., Whidborne, J.F.: Multiobjective Optimisation and Control. Research Studies Press, Baldock (2002)

    Google Scholar 

  23. Schütze, O., Lara, A., Coello Coello, C.A.: The directed search method for unconstrained multi-objective optimization problems. In: Proceedings of the EVOLVE – A Bridge Between Probability, Set Oriented Numerics, and Evolutionary Computation (2011)

    Google Scholar 

  24. Lara, A., Alvarado, S., Salomon, S., Avigad, G., Coello Coello, C.A., Schütze, O.: The gradient free directed search method as local search within multi-objective evolutionary algorithms. In: Schütze, O., Coello Coello, C.A., Tantar, A.-A., Tantar, E., Bouvry, P., Del Moral, P., Legrand, P. (eds.) EVOLVE - A Bridge Between Probability, Set Oriented Numerics, and Evolutionary Computation II. AISC, vol. 175, pp. 153–168. Springer, Heidelberg (2012)

    Chapter  Google Scholar 

  25. Nocedal, J., Wright, S.: Numerical Optimization. Springer Series in Operations Research and Financial Engineering. Springer (2006)

    Google Scholar 

  26. Das, I., Dennis, J.: Normal-boundary intersection: A new method for generating the Pareto surface in nonlinear multicriteria optimization problems. SIAM Journal of Optimization 8, 631–657 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  27. Schäffler, S., Schultz, R., Weinzierl, K.: A stochastic method for the solution of unconstrained vector opimization problems. Journal of Optimization, Theory and Application 114(1), 209–222 (2002)

    Article  MATH  Google Scholar 

  28. Liu, G.P., Daley, S.: Optimal-tuning PID controller design in the frequency domain with application to a rotary hydraulic system. Control Engineering Practice 7(7), 821–830 (1999)

    Article  Google Scholar 

  29. Liu, G.P., Daley, S.: Optimal-tuning nonlinear PID control of hydraulic systems. Control Engineering Practice 8(9), 1045–1053 (2000)

    Article  Google Scholar 

  30. Panda, S.: Multi-objective PID controller tuning for a FACTS-based damping stabilizer using non-dominated sorting genetic algorithm-II. International Journal of Electrical Power & Energy Systems 33(7), 1296–1308 (2011)

    Article  Google Scholar 

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Author information

Authors and Affiliations

  1. School of Engineering, University of California at Merced, Merced, CA, 95344, USA

    Yousef Naranjani & Jian-Qiao Sun

  2. Department of Mechanics, Tianjin University, Tianjin, 300072, China

    Fu-Rui Xiong

  3. Depto de Computacion, CINVESTAV-IPN, Mexico City, 07360, Mexico

    Carlos Hernández & Oliver Schütze

Authors
  1. Yousef Naranjani
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  2. Carlos Hernández
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  3. Fu-Rui Xiong
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  4. Oliver Schütze
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  5. Jian-Qiao Sun
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Corresponding author

Correspondence to Yousef Naranjani .

Editor information

Editors and Affiliations

  1. , Leiden Institute of Advanced, Leiden University, Niels Bohrweg 1, Leiden, 2333CA, Netherlands

    Michael Emmerich

  2. , Leiden Institute of Advanced, Leiden University, Niels Bohrweg 1, Leiden, 2333CA, Netherlands

    Andre Deutz

  3. , Depto. de Computacion, CINVESTAV-IPN, Av. IPN No. 2508, Col. San Pedro Zacatenco, Mexico, 07360, Mexico

    Oliver Schuetze

  4. , Leiden Institute of Advanced, Leiden University, Niels Bohrweg 1, Leiden, 2333CA, Netherlands

    Thomas Bäck

  5. , Luxembourg Centre for, University of Luxembourg, Avenue des Hauts Fourneaux 7, Belval, 4362, Luxembourg

    Emilia Tantar

  6. , Computer Science and, University of Luxembourg, rue Richard Coudenhove-Kalergi 6, Luxembourg, 1359, Luxembourg

    Alexandru-Adrian Tantar

  7. , Bordeaux Mathematical Institute, Université Bordeaux I, cours de la Libération 351, Talence cedex, 33405, France

    Pierre Del Moral

  8. , UFR Sciences et Modélisation, Université Bordeaux Segalen, 3ter place de la Victoire, Bordeaux, 33076, France

    Pierrick Legrand

  9. Faculty of Sci., Tech. & Communication, Computer Science and, University of Luxembourg, rue Richard Coudenhove-Kalergi 6, Luxembourg, 1359, Luxembourg

    Pascal Bouvry

  10. , Depto. de Computatción, CINVESTAV-IPN, Av. IPN No. 2508, Col. San Pedro Zacatenco, Mexico, 07360, Mexico

    Carlos A. Coello

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Naranjani, Y., Hernández, C., Xiong, FR., Schütze, O., Sun, JQ. (2013). A Hybrid Algorithm for the Simple Cell Mapping Method in Multi-objective Optimization. In: Emmerich, M., et al. EVOLVE - A Bridge between Probability, Set Oriented Numerics, and Evolutionary Computation IV. Advances in Intelligent Systems and Computing, vol 227. Springer, Heidelberg. https://doi.org/10.1007/978-3-319-01128-8_14

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  • DOI: https://doi.org/10.1007/978-3-319-01128-8_14

  • Publisher Name: Springer, Heidelberg

  • Print ISBN: 978-3-319-01127-1

  • Online ISBN: 978-3-319-01128-8

  • eBook Packages: EngineeringEngineering (R0)

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