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Solving multiobjective random interval programming problems by a capable neural network framework

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Abstract

In this paper, the stability of a class of nonlinear control systems is analyzed. We first construct an optimal control problem by inserting a suitable performance index, which this problem is referred to as an infinite horizon problem. By a suitable change of variable, the infinite horizon problem is reduced to a finite horizon problem. We then present a feedback controller designing approach for the obtained finite horizon control problem. This approach involves a neural network scheme for solving the nonlinear Hamilton Jacobi Bellman (HJB) equation. By using the neural network method, an analytic approximate solution for value function and suboptimal feedback control law is achieved. A learning algorithm based on a dynamic optimization scheme with stability and convergence properties is also provided. Some illustrative examples are employed to demonstrate the accuracy and efficiency of the proposed plan. As a real life application in engineering, the stabilization of a micro electro mechanical system is studied.

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References

  1. Sakawa M, Nishizaki I, Katagiri H (2011) Fuzzy stochastic multi-objective programming. Springer, Berlin

    Book  MATH  Google Scholar 

  2. Kall P, Mayer J (2004) Stochastic linear programming, Springer, Berlin

  3. Dempster AP (1968) Upper and lower probabilities generated by a random closed interval. Ann Math Statist 39:957–966

    Article  MathSciNet  MATH  Google Scholar 

  4. Dubois D, Prade H (1987) The mean value of a fuzzy number. Fuzzy Set Syst 24(3):279–300

    Article  MathSciNet  MATH  Google Scholar 

  5. Lu HW, Cao MF, Li J, Huang GH, He L (2015) An inexact programming approach for urban electric power systems management under random-interval-parameter uncertainty. Appl Math Model 39:1757–1768

    Article  MathSciNet  Google Scholar 

  6. Ferson S, Ginzburg L, Kreinovich V, Longpré L , Aviles M (2005) Exact bounds on finite populations of interval data. Reliab Comput 11(3):207–233

    Article  MathSciNet  MATH  Google Scholar 

  7. Kreinovich V, Nguyen HT, Wu B (2007) On-line algorithms for computing mean and variance of interval data, and their use in intelligent systems. Inform Sci 177:3228–3238

    Article  MathSciNet  MATH  Google Scholar 

  8. Ferson S, Ginzburg L, Kreinovich V, Longpré L., Aviles M (2002) Computing variance for interval data is NP-hard. ACM SIGACT News 33(2):108–118

    Article  Google Scholar 

  9. Hansen E (1997) Sharpness in interval computations. Reliab Comput 3(1):17–29

    Article  MathSciNet  MATH  Google Scholar 

  10. Kearfott RB (1996) Rigorous global search: continuous problems. Kluwer Academic Publishers, Dordrecht

    Book  MATH  Google Scholar 

  11. Arjmandzadeh Z, Safi MR, Nazemi A (2017) A new neural network model for solving random interval linear programming problems. Neural Netw 89:11–18

    Article  Google Scholar 

  12. Arjmandzadeh Z, Safi MR A goal programming approach for solving random interval linear programming problem., Turkish Journal of Mathematics. https://doi.org/10.3906/mat-1509-65

  13. Arjmandzadeh Z, Safi MR (2018) A generalisation of variance model for solving random interval linear programming problems. Int J Syst Sci: Oper Logist 5(1):16–24. https://doi.org/10.1080/23302674.2016.1224951

    Google Scholar 

  14. Barik SK, Biswal MP, Chakravarty D (2012) Two-stage stochastic programming problems involving interval discrete random variables. OPSEARCH 49:280–298

    Article  MathSciNet  MATH  Google Scholar 

  15. Tank DW, Hopfield JJ (1986) Simple neural optimization networks: an A/D converter, signal decision circuit, and a linear programming circuit. IEEE Tran Circuits Syst 33:533–541

    Article  Google Scholar 

  16. Ding K, Huang N-J (2008) A new class of interval projection neural networks for solving interval quadratic program. IEEE Chaos Solitons and Fractals 35:718–725

    Article  MATH  Google Scholar 

  17. Sabouri J, Effati KS, Pakdaman M (2016) A neural network approach for solving a class of fractional optimal control problems., Neural Processing Letter. to appear

  18. Nazemi AR (2012) A dynamic system model for solving convex nonlinear optimization problems. Commun Nonlinear Sci Numer Simul 17:1696–1705

    Article  MathSciNet  MATH  Google Scholar 

  19. Nazemi AR (2011) A dynamical model for solving degenerate quadratic minimax problems with constraints. J Comput Appl Math 236:1282–1295

    Article  MathSciNet  MATH  Google Scholar 

  20. Xia Y, Feng G, Wang J (2004) A recurrent neural network with exponential convergence for solving convex quadratic program and related linear piecewise equation. Neural Netw 17:1003– 1015

    Article  MATH  Google Scholar 

  21. Xia Y, Feng G (2005) An improved network for convex quadratic optimization with application to real-time beamforming. Neurocomputing 64:359–374

    Article  Google Scholar 

  22. Xia Y (1996) A new neural network for solving linear and quadratic programming problems. IEEE Trans Neural Netw 7:1544–1547

    Article  Google Scholar 

  23. Xia Y, Wang J (2004) A general projection neural network for solving monotone variational inequality and related optimization problems. IEEE Trans Neural Netw 15:318–328

    Article  Google Scholar 

  24. Xia Y, Wang J (2004) A recurrent neural network for nonlinear convex optimization subject to nonlinear inequality constraints. IEEE Trans Neural Netw Circuits Syst 51:447–458

    MathSciNet  MATH  Google Scholar 

  25. Xue X, Bian W (2007) A project neural network for solving degenerate convex quadratic program. Neurocomputing 70:2449–2459

    Article  Google Scholar 

  26. Xue X, Bian W (2009) A project neural network for solving degenerate quadratic minimax problem with linear constraints. Neurocomputing 72:1826–1838

    Article  Google Scholar 

  27. Yang Y, Cao J (2006) A delayed neural network method for solving convex optimization problems. Int J Neural Syst 16:295–303

    Article  Google Scholar 

  28. Yang Y, Cao J (2008) A feedback neural network for solving convex constraint optimization problems. Appl Math Comput 201:340–350

    MathSciNet  MATH  Google Scholar 

  29. Miranda E, Couso I, Gil P (2005) Random intervals as a model for imprecise information. Fuzzy Set Syst 154:386–412

    Article  MathSciNet  MATH  Google Scholar 

  30. Geoffrion AM (1967) Stochastic programming with aspiration or fractile criteria. Manag Sci 13:672–679

    Article  MathSciNet  MATH  Google Scholar 

  31. Faraut J, Konyi A (1994) Analysis on symmetric cones. In: Oxford Mathematical Monographs. Oxford University Press, New York

  32. Benson HY, Vanderbei RJ (2003) Solving problems with semidefinite and related constraints using interior-point methods for nonlinear programming. Math Programming Ser B 95(2):279– 302

    Article  MathSciNet  MATH  Google Scholar 

  33. Bazaraa MS, Sherali HD, Shetty CM (1993) Nonlinear programming-theory and algorithms, 2nd edn. Wiley, New York

    MATH  Google Scholar 

  34. Miller RK, Michel AN (1982) Ordinary differential equations. Academic Press, New York

    MATH  Google Scholar 

  35. Quarteroni A, Sacco R, Saleri F (2007) Numerical mathematics, texts in applied mathematics, 2nd edn. Springer, Berlin, p 37

    MATH  Google Scholar 

  36. Fukushima M (1992) Equivalent differentiable optimization problems and descent methods for asymmetric variational inequality problems. Math Programm 53(1–3):92–110

    MathSciNet  MATH  Google Scholar 

  37. Amann H (1990) Ordinary differential equations: An introduction to nonlinear analysis. Walter de Gruyter, Berlin

    Book  MATH  Google Scholar 

  38. Grant M, Boyd S, Ye Y (2008) CVX: Matlab software for disciplined convex programming

  39. Kalyanmoy D (2014) Multi-objective optimization. Search methodologies. Springer, Boston, pp 403–449

    Google Scholar 

  40. Kalyanmoy D, Padhye N (2014) Enhancing performance of particle swarm optimization through an algorithmic link with genetic algorithms. Comput Optim Appl 57(3):761–794

    Article  MathSciNet  MATH  Google Scholar 

  41. Padhye N, Piyush B, Kalyanmoy D (2013) Improving differential evolution through a unified approach. J Glob Optim 55(4):771–799

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgments

The authors are very grateful to the editor and anonymous reviewers for their suggestions in improving the quality of the paper.

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Authors and Affiliations

  1. Department of Mathematics, Semnan University, Semnan, Iran

    Ziba Arjmandzadeh & Mohammadreza Safi

  2. Faculty of Mathematical Sciences, Shahrood University of Technology, P.O. Box 3619995161-316, Shahrood, Iran

    Alireza Nazemi

Authors
  1. Ziba Arjmandzadeh
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  2. Alireza Nazemi
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  3. Mohammadreza Safi
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Corresponding author

Correspondence to Alireza Nazemi.

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Arjmandzadeh, Z., Nazemi, A. & Safi, M. Solving multiobjective random interval programming problems by a capable neural network framework. Appl Intell 49, 1566–1579 (2019). https://doi.org/10.1007/s10489-018-1344-6

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  • DOI: https://doi.org/10.1007/s10489-018-1344-6

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