Optimization problems with convex epigraphs. Application to optimal control

Arkadii Kryazhimskii

International Journal of Applied Mathematics and Computer Science (2001)

  • Volume: 11, Issue: 4, page 773-801
  • ISSN: 1641-876X

Abstract

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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.

How to cite

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Kryazhimskii, Arkadii. "Optimization problems with convex epigraphs. Application to optimal control." International Journal of Applied Mathematics and Computer Science 11.4 (2001): 773-801. <http://eudml.org/doc/207531>.

@article{Kryazhimskii2001,
abstract = {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.},
author = {Kryazhimskii, Arkadii},
journal = {International Journal of Applied Mathematics and Computer Science},
keywords = {nonconvex optimization; global optimization methods; lower semicontinuous function; extended convex optimization problem; optimal control},
language = {eng},
number = {4},
pages = {773-801},
title = {Optimization problems with convex epigraphs. Application to optimal control},
url = {http://eudml.org/doc/207531},
volume = {11},
year = {2001},
}

TY - JOUR
AU - Kryazhimskii, Arkadii
TI - Optimization problems with convex epigraphs. Application to optimal control
JO - International Journal of Applied Mathematics and Computer Science
PY - 2001
VL - 11
IS - 4
SP - 773
EP - 801
AB - 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.
LA - eng
KW - nonconvex optimization; global optimization methods; lower semicontinuous function; extended convex optimization problem; optimal control
UR - http://eudml.org/doc/207531
ER -

References

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  7. Kryazhimskii A.V. (1999): Convex optimization via feedbacks. — SIAM J. Contr. Optim., Vol.37, No.1, pp.278–302. Zbl0917.90256
  8. 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. Zbl0965.49020
  9. 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). 
  10. Pontryagin L.S., Boltyanskii V.G., Gamkrelidze R.V. and Mishchenko E.F. (1969): Mathematical Theory of Control Processes. — Moscow: Nauka, (in Russian). 
  11. 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. 
  12. Vasiliev F.P. (1981): Solution Methods for Extremal Problems. — Moscow, Nauka, (in Russian). 
  13. Warga J. (1975): Optimal Control of Differential and Functional Equations. — New York: Academic Press. Zbl0272.49005
  14. Zangwill W.I. and Garcia C.B. (1981): Pathways to Solutions, Fixed Points and Equilibria. — Englewood Cliffs: Prentice-Hall. Zbl0512.90070

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