# Optimization problems with convex epigraphs. Application to optimal control

International Journal of Applied Mathematics and Computer Science (2001)

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

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topKryazhimskii, 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

top- Bertsekas D.P. (1982): Constrained Optimization and Lagrange Multiplier Methods. — New York: Academic Press. Zbl0572.90067
- 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. Zbl0871.90063
- Fedorenko R.P. (1978): Approximate Solution of Optimal Control Problems. — Moscow: Nauka, (in Russian). Zbl0462.49001
- Gabasov R., Kirillova F.M. and Tyatyushkin A.I. (1984): Constructive Optimization Problems, Part 1: Linear Problems. — Minsk: Universitetskoye, (in Russian). Zbl0581.90051
- Krasovskii N.N. (1985): Control of Dynamical Systems. — Moscow: Nauka, (in Russian).
- Krasovskii N.N. and Subbotin A.I. (1988): Game-Theoretical Control Problems. — Berlin: Springer.
- Kryazhimskii A.V. (1999): Convex optimization via feedbacks. — SIAM J. Contr. Optim., Vol.37, No.1, pp.278–302. Zbl0917.90256
- 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
- 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).
- Pontryagin L.S., Boltyanskii V.G., Gamkrelidze R.V. and Mishchenko E.F. (1969): Mathematical Theory of Control Processes. — Moscow: Nauka, (in Russian).
- 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.
- Vasiliev F.P. (1981): Solution Methods for Extremal Problems. — Moscow, Nauka, (in Russian).
- Warga J. (1975): Optimal Control of Differential and Functional Equations. — New York: Academic Press. Zbl0272.49005
- 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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