Skip to main content
SpringerLink
Log in

Extension technology and extrema selections in a stochastic multistart algorithm for optimal control problems

  • Published:
Journal of Global Optimization Aims and scope Submit manuscript

Abstract

The paper proposes a method for finding the minimum value of a functional in nonlinear nonconvex optimal control problems. The method takes advantage of the hidden convexity property of the controlled differential equations systems. Application of the multistart idea with extrema selection procedures makes it possible to create software that does not strongly depend on the problem size and supplies additional information about the object under investigation. Three test problems are considered to show specific properties of using the stochastic multistart algorithm and extension numerical technology.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4

Similar content being viewed by others

References

  1. Bondarenko, A.S., Bortz, D.M., More J.J.: COPS: large-scale nonlinearly constrained optimization problems. Technical Memorandum ANL/MCS-TM-237, pp. 12–14 (1999)

  2. Evtushenko, Y.G., Polovinkin, M.A.: The parallel methods for solving global optimization problems. In: Proceeding of IV International Conference on Parallel computing and control problems, pp. 18–39 (2008). (in Russian)

  3. Floudas, C.A., Gounaris, C.E.: A review of recent advanced in global optimization. J. Glob. Optim. 45, 3–38 (2009)

    Article  Google Scholar 

  4. Chachuat, B., Singer, A.B., Barton, P.I.: Global methods for dynamic optimization and mixed-integer dynamic optimization. Ind. Eng. Chem. Res. 45, 8373–8392 (2006)

    Article  Google Scholar 

  5. Barton, P.I., Lee, C.K., Yunt, M.: Optimization of hybrid systems. Comput. Chem. Eng. 30, 1576–1589 (2006)

    Article  Google Scholar 

  6. Lin, Y.D., Stadtherr, M.A.: Deterministic global optimization of nonlinear dynamic systems. AIChE J. 53, 866–875 (2007)

    Article  Google Scholar 

  7. Lopez Cruz, I.L.: Efficient Evolutionary Algorithms for Optimal Control. Ph.D. thesis (2002)

  8. Moiseev, N.N.: The Elements of the Optimal Systems Theory. Nauka, Moscow (1975). (in Russian)

    MATH  Google Scholar 

  9. Bellman, R.: The Dynamical Programming. Princeton University Press, Princeton (1957)

    MATH  Google Scholar 

  10. Krotov, V.F.: Global Methods in Optimal Control Theory. Marcel Dekker Inc., New York (1996)

    MATH  Google Scholar 

  11. Strekalovsky, A.S., Yanulevich, M.V.: The global search in optimal control problem with cost terminal functional which is presented by difference of two convex functions. J. Comput. Math. Math. Phys. 48, 1187–1201 (2008). (in Russian)

    Google Scholar 

  12. Zarodnyuk, T.S., Gornov, A.Y.: A technique of finding global extremum in optimal control problems. Modern techniques. System analysis. Simulation 3, 70–76 (2008). (in Russian)

  13. Gornov, A.Y., Zarodnyuk, T.S.: The “curvilinear search” method of global extremum in optimal control problems. Modern techniques. System analysis. Simulation 3, 19–26 (2009). (in Russian)

  14. Krotov, V.F., Gurman, V.I.: The Methods and Problems of Optimal Control. Nauka, Moscow (1973). (in Russian)

    Google Scholar 

  15. Dykhta, V., Samsonyuk, O.: Some applications of Hamilton-Jacobi inequalities for classical and impulsive optimal control problems. Eur. J. Control. 17, 55–69 (2011)

    Article  MathSciNet  Google Scholar 

  16. Dykhta, V.A., Sorokin, S.P.: Hamilton-Jacobi inequalities and the optimality conditions in the problems of control with common end constraints. Autom. Remote Control. 72, 1808–1821 (2011)

    Article  MathSciNet  Google Scholar 

  17. Tolstonogov, A.A.: Differential Inclusions in a Banach Space, Mathematics and Its Applications. Kluwer Academic Publishers, Dordrecht (2000)

    Book  Google Scholar 

  18. Shary, S.P.: Randomized algorithms in interval global optimization. Numer. Anal. Appl. 1(4), 376–389 (2008)

    Article  Google Scholar 

  19. Gamkrelidze, R.V.: Principles of Optimal Control Theory. Plenum Press, New York (1978)

    Book  Google Scholar 

  20. Mordukhovich, B.S.: Approximation Methods in Optimization and Control Problems. Nauka, Moscow (1988). (in Russian)

    MATH  Google Scholar 

  21. Clarke, F.H.: Optimization and Nonsmooth Analysis. Wiley, New York (1983)

    MATH  Google Scholar 

  22. Gornov, A.Y., Zarodnyuk, T.S.: Tunneling algorithm for solving nonconvex optimal control problems. In: Chinchuluun, A., et al. (eds.) Optimization, Simulation, and Control. Springer Optimization and Its Applications, vol. 76, pp. 289–299. Springer, Berlin (2013)

    Chapter  Google Scholar 

  23. Zhigljavsky, A., Zilinskas, A.: Stochastic Global Optimization. Springer, New York (2008)

    MATH  Google Scholar 

  24. Gornov, A.Y., Zarodnyuk, T.S., Madzhara, T.I., Daneyeva, A.V., Veyalko, I.A.: A collection of test multiextremal optimal control problems. In: Chinchuluun, A., et al. (eds.) Optimization, Simulation, and Control. Springer Optimization and Its Applications, vol. 76, pp. 257–274. Springer, Berlin (2013)

    Chapter  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Matrosov Institute for Systems Dynamics and Control Theory of SB RAS, 134 Lermontov Str., Irkutsk, Russia, 664033

    Alexander Yu. Gornov, Tatiana S. Zarodnyuk, Anton S. Anikin & Evgeniya A. Finkelstein

Authors
  1. Alexander Yu. Gornov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Tatiana S. Zarodnyuk
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Anton S. Anikin
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Evgeniya A. Finkelstein
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Alexander Yu. Gornov.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

The authors are supported by RFBR Grant 17-07-00627.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Gornov, A.Y., Zarodnyuk, T.S., Anikin, A.S. et al. Extension technology and extrema selections in a stochastic multistart algorithm for optimal control problems. J Glob Optim 76, 533–543 (2020). https://doi.org/10.1007/s10898-019-00821-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10898-019-00821-x

Keywords

Search

Navigation