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Variational Control Problems with Constraints Via Exact Penalization

  • V. F. Demyanov
  • F. Giannessi
  • G. Sh. Tamasyan
Chapter
Part of the Nonconvex Optimization and Its Applications book series (NOIA, volume 79)

Abstract

The Exact Penalization approach to solving constrained problems of Calculus of Variations described in [17] is extended to the case of variational problems where the functional and the constraints contain a control function. The constraints are of both the equality- and inequality-type constraints. The initial constrained problem is reduced to an unconstrained one. The related unconstrained problem is essentially nonsmooth. Necessary optimality conditions are derived.

Key words

Calculus of Variations Equality- and Inequality-type Constraints Necessary optimality conditions Penalty Function Exact Penalty Function Nonsmooth Analysis 

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References

  1. [1]
    Bliss G.A. (1946), Lectures on the Calculus of Variations. Chicago, Univ. of Chicago Press.zbMATHGoogle Scholar
  2. [2]
    Boukary D., Fiacco A.V. (1995), Survey of penalty, exact-penalty and multiplier methods from 1968 to 1993. Optimization, Vol. 32, No. 4, pp. 301–334.MathSciNetGoogle Scholar
  3. [3]
    Demyanov V.F., Vasiliev L.V. (1985) Nondifferentiable optimization. New-York, Springer-Optimization Software.zbMATHGoogle Scholar
  4. [4]
    Demyanov V.F. (1992), Nonsmooth problems in Calculus of Variations. In Lecture Notes in Economics and Math. Systems, v. 382. Advances in Optimization. Eds. W. Oettli, D. Pallaschke, pp. 227–238. Berlin, Springer Verlag.Google Scholar
  5. [5]
    Demyanov V.F. (1992a), Calculus of Variations in nonsmooth presentation. In Nonsmooth Optimization: Methods and Applications. Ed. F. Giannessi, pp. 76–91. Singapore, Gordon and Breach.Google Scholar
  6. [6]
    Demyanov V.F. (1994), Exact penalty functions in nonsmooth optimization problems. Vestnik of St. Petersburg University, ser. 1, issue 4 (No. 22), pp. 21–27.Google Scholar
  7. [7]
    Demyanov V.F. (2000), Conditions for an extremum and variational problems. St. Petersburg, St. Petersburg University Press.Google Scholar
  8. [8]
    Demyanov V.F. (2003) Constrained problems of Calculus of Variations via Penalization Technique. In: Equilibrium Problems and Variational Models. A. Maugeri, F. Giannessi (Eds.), pp. 79–108. Kluwer Academic Publishers.Google Scholar
  9. [9]
    Demyanov V.F., Di Pillo G., Facchinei F. (1998), Exact penalization via Dini and Hadamard conditional derivatives. Optimization Methods and Software, Vol. 9, pp. 19–36.zbMATHMathSciNetGoogle Scholar
  10. [10]
    Demyanov V.F., Giannessi F., and Karelin V.V. (1998), Optimal Control Problems via Exact Penalty Functions. Journal of Global Optimization, Vol. 12, No. 3, pp.215–223.CrossRefMathSciNetGoogle Scholar
  11. [11]
    Demyanov V.F., Rubinov A.M. (1995), Constructive Nonsmooth Analysis. Frankfurt a/M., Peter Lang Verlag.zbMATHGoogle Scholar
  12. [12]
    Di Pillo G., Facchinei F. (1989), Exact penalty functions for nondifferentiable programming problems. In Nonsmooth Optimization and Related Topics. Eds. F.H. Clarke, V.F. Demyanov and F. Giannessi, pp. 89–107. New York, Plenum.Google Scholar
  13. [13]
    Eremin I.I. (1966), On the penalty method in convex programming. In Abstracts of ICM-66. Section 14., Moscow, 1966.Google Scholar
  14. [14]
    Eremin I.I. (1967). A method of “penalties” in Convex Programming. Soviet Mathematics Doklady (4), 748–751.MathSciNetGoogle Scholar
  15. [15]
    Fletcher R. (1983), Penalty functions. In Mathematical programming: the state of the art. (Eds. A. Bachen, M. Grötschel, B. Korte), Springer-Verlag, Berlin, pp. 87–114.Google Scholar
  16. [16]
    Han S., Mangasarian O. (1979), Exact penalty functions in nonlinear programming. Mathematical Programming, Vol.17, pp. 251–269.zbMATHCrossRefMathSciNetGoogle Scholar
  17. [17]
    Hestenes M.R. (1966), Calculus of Variations and Optimal Control Theory. New York, John Wiley & Sons.zbMATHGoogle Scholar
  18. [18]
    Kaplan A.A., Tichatschke R. (1994), Stable methods for ill-posed variational problems. Berlin, Akademie Verlag.zbMATHGoogle Scholar
  19. [19]
    Pontryagin L.S., Boltyanskii V.G., Gamkrelidze R.V., and Mishchenko E.F. (1962), The Mathematical theory of Optimal Processes. New York, Interscience Publishers.zbMATHGoogle Scholar
  20. [20]
    Rockafellar R.T. (1970) Convex Analysis. Princeton. N.J., Princeton University Press.zbMATHGoogle Scholar
  21. [21]
    Zangwill W.L. (1967), Nonlinear programming via penalty functions. Management Science, Vol. 13, pp. 344–358.MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2005

Authors and Affiliations

  • V. F. Demyanov
    • 1
  • F. Giannessi
    • 2
  • G. Sh. Tamasyan
    • 1
  1. 1.Applied Mathematics Dept.St. Petersburg State UniversitySt. PetersburgRussia
  2. 2.Dept. of MathematicsUniversity of PisaPisaItaly

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