Optimization problems with convex epigraphs. Application to optimal control

• Autorzy: Kryazhimskii A. V.
• Języki publikacji: EN

Abstrakty

EN

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.

Słowa kluczowe

EN

nonconvex optimization; global optimization methods; optimal control

PL

optymalizacja niegładka; metoda optymalizacji globalnej; sterowanie optymalne

• Wydawca: Oficyna Wydawnicza Uniwersytetu Zielonogórskiego
• Czasopismo: International Journal of Applied Mathematics and Computer Science
• Rocznik: 2001
• Tom: Vol. 11, no. 4
• Strony: 773--801
• Opis fizyczny: Bibliogr. 14 poz.

Twórcy

autor

Kryazhimskii A. V.

• V.A. Steklov Institute of Mathematics Russian Academy of Sciences Gubkin Str. 8, 117966 Moscow, Russia,

Bibliografia

• [1] Bertsekas D.P. (1982): Constrained Optimization and Lagrange Multiplier Methods. - New York: Academic Press.
• [2] 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.
• [3] Fedorenko R.P. (1978): Approximate Solution of Optimal Control Problems. - Moscow: Nauka, (in Russian).
• [4] Gabasov R., Kirillova F.M. and Tyatyushkin A.I. (1984): Constructive Optimization Problems, Part 1: Linear Problems. - Minsk: Universitetskoye, (in Russian).
• [5] Krasovskii N.N. (1985): Control of Dynamical Systems. - Moscow: Nauka, (in Russian).
• [6] Krasovskii N.N. and Subbotin A.I. (1988): Game-Theoretical Control Problems. - Berlin: Springer.
• [7] Kryazhimskii A.V. (1999): Convex optimization via feedbacks. - SIAM J. Contr. Optim., Vol. 37, No. 1, pp. 278-302.
• [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.
• [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.
• [14] Zangwill W.I. and Garcia C.B. (1981): Pathways to Solutions, Fixed Points and Equilibria. - Englewood Cliffs: Prentice-Hall.

Uwagi

The work was supported by the Russian Foundation for Basic Research, Project No. 00-01-00682, and the International Institute for Applied Systems Analysis, Laxenburg, Austria.