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On global search in nonconvex optimal control problems

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Abstract

Optimal control problems with a nonconvex quadratic functional of Lagrange are considered. On the base of global optimality conditions we develop a global search algorithm, one of the principal module of which is represented by special local search method. The results of computational testing presented.

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References

  1. Pontryagin, L.S., Boltyanskii, V.G., Gamkrelidze, R.V., Mishchenko, E.F.: The Mathematical Theory of Optimal Processes. Interscience, New York (1976)

    Google Scholar 

  2. Chernousko, F.L., Ananievski, I.M., Reshmin, S.A.: Control of Nonlinear Dynamical Systems: Methods and Applications. Springer, Berlin (2008)

    Book  MATH  Google Scholar 

  3. Chernousko, F.L.: State Estimation for Dynamic Systems. CRC Press, Boca Raton (1994)

    MATH  Google Scholar 

  4. Vasil’ev, F.P.: Optimization Methods. Factorial Press, Moscow (2002). (in Russian)

    Google Scholar 

  5. Gabasov, R., Kirillova, F.M.: Optimization of Linear Systems. Plenum Press, New York (1979)

    MATH  Google Scholar 

  6. Vasiliev, O.V.: Optimization Methods. Word Federation Publishing Company, Atlanta (1996)

    MATH  Google Scholar 

  7. Pang, J.-S.: Three modelling paradigms in mathematical programming. Math. Program. Ser. B 125, 297–323 (2010)

    Article  MATH  Google Scholar 

  8. Nocedal, J., Wright, St: Numerical Optimization, 2nd edn. Springer, New York (2006)

    MATH  Google Scholar 

  9. Srochko, V.A.: Iterative Solution of Optimal Control Problems. Fizmatlit, Moscow (2000). (in Russian)

    Google Scholar 

  10. Strekalovsky, A.S.: Elements of Nonconvex Optimization. Nauka, Novosibirsk (2003). (in Russian)

    Google Scholar 

  11. Alexandrov, A.D.: On surfaces represented a difference of convex functions. In: Proceedings of Academy of Sciences of KSSR. Ser. Math. Mech. 3, 3–20 (1949) (in Russian)

  12. Alexandrov, A.D.: The surfaces that can be represented by a difference of convex functions. In: Proceedings of Academy of Sciences of USSR. Ser. Math. Mech. 72(4), 613–616 (1950)

  13. Strekalovsky, A.S.: Global optimality conditions for optimal control problems with functions of A.D. Alexandrov. J. Optim. Theory Appl. 159, 297–321 (2013)

  14. Strekalovsky, A.S.: Maximizing a state convex Lagrange functional in optimal control. Autom. Remote Control 73(6), 949–961 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  15. Strekalovsky, A.S.: Optimal control problems with terminal functionals represented as a difference of two convex functions. Comput. Math. Math. Phys. 47(11), 1788–1801 (2007)

    Article  MathSciNet  Google Scholar 

  16. Clarke, F.: Optimization and Nonsmooth Analysis, 2nd edn. SIAM, Philadelphia (1990)

    Book  MATH  Google Scholar 

  17. Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation. I: Basic Theory. II: Applications. Springer, Berlin (2006)

    Google Scholar 

  18. Trefethen, L.N., Bau, D.: Numer. Linear Algebra. SIAM, Philadelphia (1997)

    Book  MATH  Google Scholar 

  19. Strekalovsky, A.S.: On global maximum of a convex terminal functional in optimal control problems. J. Glob. Optim. 7, 75–91 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  20. Phedorenko, R.P.: Approximate Solution of Optimal Control Problems. Nauka, Moskow (1978). (in Russian)

    Google Scholar 

  21. Vicente, L.N., Calamai, P.H., Judice, J.J.: Generation of disjointly constrained bilinear programming test problems. Comput. Optim. Appl. 1(3), 299–306 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  22. Calamai, P.H., Vicente, L.N.: Generating quadratic bilevel programming test problems. ACM Trans. Math. Softw. 20, 103–119 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  23. Strekalovsky, A.S., Yanulevich, M.V.: Global search in a nonconvex optimal control problem. J. Comput. Syst. Sci. Int. 52(6), 30–45 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  24. Hiriart-Urruty, J.-B., Lemarshal, C.: Convex Analysis and Minimization Algorithms. Springer, Berlin (1993)

    Google Scholar 

  25. Hiriart-Urruty, J.-B.: Generalized differentiability, duality and optimization for problem dealing with difference of convex functions. In: Ponstein, J. (ed.) Convexity and Duality in Optimization, pp. 37–69. Springer, Berlin (1985)

    Chapter  Google Scholar 

  26. Tuy, H.: D.C. optimization: theory, methods and algorithms. In: Horst, R., Pardalos, P.M. (eds.) Handbook of Global Optimization, pp. 149–216. Kluwer Academic Publishers, Dordrecht (1995)

    Chapter  Google Scholar 

  27. Strekalovsky, A.S., Yanulevich, M.V.: Global search in the optimal control problem with a therminal objective functional represented as a difference of two convex functions. Comput. Math. Math. Phys. 48(7), 1119–1132 (2008)

    Article  MathSciNet  Google Scholar 

  28. Strekalovsky, A.S., Yanulevich, M.V.: On solving nonconvex optimal control problems with a terminal objective functional. Numer. Methods Program. 11, 269–280 (2010). (in Russian)

    Google Scholar 

  29. Strekalovsky, A.S.: Local search for nonconvex optimal control problems of Bolza. Numer. Methods Program. 11, 344–350 (2010)

    Google Scholar 

  30. Strekalovsky, A.S.: On solving optimization problems with hidden nonconvex structures. In: Themistocles, M., Floudas, Ch. A., Butenko, S. (eds.) Optimization in Science and Engineering, pp. 465–502. Springer, New York (2014)

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Correspondence to Maxim V. Yanulevich.

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Strekalovsky, A.S., Yanulevich, M.V. On global search in nonconvex optimal control problems. J Glob Optim 65, 119–135 (2016). https://doi.org/10.1007/s10898-015-0321-4

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