Optimization problems with convex epigraphs. Application to optimal control
Kryazhimskii, Arkadii
International Journal of Applied Mathematics and Computer Science, Tome 11 (2001), p. 773-801 / Harvested from The Polish Digital Mathematics Library
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For a class of infinite-dimensional minimization problems with nonlinear equality constraints, an iterative algorithm for finding global solutions is suggested. A key assumption is the convexity of the ''epigraph'', a set in the product of the image spaces of the constraint and objective functions. A convexification method involving randomization is used. The algorithm is based on the extremal shift control principle due to N.N. Krasovskii. An application to a problem of optimal control for a bilinear control system is described.

Publié le : 2001-01-01
EUDML-ID : urn:eudml:doc:207531 
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     title = {Optimization problems with convex epigraphs. Application to optimal control},
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     year = {2001},
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Kryazhimskii, Arkadii. Optimization problems with convex epigraphs. Application to optimal control. International Journal of Applied Mathematics and Computer Science, Tome 11 (2001) pp. 773-801. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-amcv11i4p773bwm/

[000] Bertsekas D.P. (1982): Constrained Optimization and Lagrange Multiplier Methods. — New York: Academic Press. | Zbl 0572.90067

[001] Ermoliev Yu.M., Kryazhimskii A.V. and Ruszczyński A. (1997): Constraint aggregation principle in convex optimization. — Math. Programming, Series B, Vol.76, No.3, pp.353–372. | Zbl 0871.90063

[002] Fedorenko R.P. (1978): Approximate Solution of Optimal Control Problems. — Moscow: Nauka, (in Russian). | Zbl 0462.49001

[003] Gabasov R., Kirillova F.M. and Tyatyushkin A.I. (1984): Constructive Optimization Problems, Part 1: Linear Problems. — Minsk: Universitetskoye, (in Russian). | Zbl 0581.90051

[004] Krasovskii N.N. (1985): Control of Dynamical Systems. — Moscow: Nauka, (in Russian).

[005] Krasovskii N.N. and Subbotin A.I. (1988): Game-Theoretical Control Problems. — Berlin: Springer.

[006] Kryazhimskii A.V. (1999): Convex optimization via feedbacks. — SIAM J. Contr. Optim., Vol.37, No.1, pp.278–302. | Zbl 0917.90256

[007] Kryazhimskii A.V. and Maksimov V.I. (1998): An iterative procedure for solving control problem with phase constraints. — Comp. Math. Math. Phys., Vol.38, No.9, pp.1423– 1428. | Zbl 0965.49020

[008] Matveyev A.S. and Yakubovich V.A. (1998): Nonconvex problems of global optimization in control theory. — Modern Mathematics and Its Applications, All-Russian Institute for Scientific and Technical Information, Vol.60, pp.128–175, (in Russian).

[009] Pontryagin L.S., Boltyanskii V.G., Gamkrelidze R.V. and Mishchenko E.F. (1969): Mathematical Theory of Control Processes. — Moscow: Nauka, (in Russian).

[010] Sonnevend G. (1986): An “analytic center” for polyhedrons and new classes of global algorithms for linear (smooth convex) programming. — Proc. 12th Conf. System Modelling and Optimization, Budapest, 1985, Berlin: Springer, LNCIS, Vol.84, pp.866–876.

[011] Vasiliev F.P. (1981): Solution Methods for Extremal Problems. — Moscow, Nauka, (in Russian).

[012] Warga J. (1975): Optimal Control of Differential and Functional Equations. — New York: Academic Press. | Zbl 0272.49005

[013] Zangwill W.I. and Garcia C.B. (1981): Pathways to Solutions, Fixed Points and Equilibria. — Englewood Cliffs: Prentice-Hall. | Zbl 0512.90070