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.
The work was supported by the Russian Foundation for Fundamental Studies (RFFI) under Grant No 03-01-00668.
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References
Bliss G.A. (1946), Lectures on the Calculus of Variations. Chicago, Univ. of Chicago Press.
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.
Demyanov V.F., Vasiliev L.V. (1985) Nondifferentiable optimization. New-York, Springer-Optimization Software.
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.
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.
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.
Demyanov V.F. (2000), Conditions for an extremum and variational problems. St. Petersburg, St. Petersburg University Press.
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.
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.
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.
Demyanov V.F., Rubinov A.M. (1995), Constructive Nonsmooth Analysis. Frankfurt a/M., Peter Lang Verlag.
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.
Eremin I.I. (1966), On the penalty method in convex programming. In Abstracts of ICM-66. Section 14., Moscow, 1966.
Eremin I.I. (1967). A method of “penalties” in Convex Programming. Soviet Mathematics Doklady (4), 748–751.
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.
Han S., Mangasarian O. (1979), Exact penalty functions in nonlinear programming. Mathematical Programming, Vol.17, pp. 251–269.
Hestenes M.R. (1966), Calculus of Variations and Optimal Control Theory. New York, John Wiley & Sons.
Kaplan A.A., Tichatschke R. (1994), Stable methods for ill-posed variational problems. Berlin, Akademie Verlag.
Pontryagin L.S., Boltyanskii V.G., Gamkrelidze R.V., and Mishchenko E.F. (1962), The Mathematical theory of Optimal Processes. New York, Interscience Publishers.
Rockafellar R.T. (1970) Convex Analysis. Princeton. N.J., Princeton University Press.
Zangwill W.L. (1967), Nonlinear programming via penalty functions. Management Science, Vol. 13, pp. 344–358.
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Demyanov, V.F., Giannessi, F., Tamasyan, G.S. (2005). Variational Control Problems with Constraints Via Exact Penalization. In: Giannessi, F., Maugeri, A. (eds) Variational Analysis and Applications. Nonconvex Optimization and Its Applications, vol 79. Springer, Boston, MA. https://doi.org/10.1007/0-387-24276-7_21
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DOI: https://doi.org/10.1007/0-387-24276-7_21
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