Metric regularity under approximations

• Autorzy: Dontchev A. L. , Veliov V. M.
• Języki publikacji: EN

Abstrakty

EN

In this paper we show that metric regularity and strong metric regularity of a set-valued mapping imply convergence of inexact iterative methods for solving a generalized equation associated with this mapping. To accomplish this, we first focus on the question how these properties are preserved under changes of the mapping and the reference point. As an application, we consider discrete approximations in optimal control.

Słowa kluczowe

EN

metric regularity; inexact iterative methods; Newton method; proximal point method; discrete approximation; optimal control

• Wydawca: Systems Research Institute, Polish Academy of Sciences
• Czasopismo: Control and Cybernetics
• Rocznik: 2009
• Tom: Vol. 38, no 4B
• Strony: 1283--1303
• Opis fizyczny: Bibliogr. 11 poz.

Twórcy

autor

Dontchev A. L.

autor

Veliov V. M.

• Mathematical Reviews, AMS, Ann Arbor, MI, USA,

Bibliografia

• DONTCHEV, A.L.(1996) An a priori estimate for discrete approximations in nonlinear optimal control. SIAM J. Control Optim. 34, 1315-1328.
• DONTCHEV, A.L. and HAGER, W.W. (1994) An inverse mapping theorem for set-valued maps. Proc. Amer. Math. Soc. 121, 481-489.
• DONTCHEV, A.L., HAGER, W.W. and VELIOV, V.M. (2000) Uniform convergence and mesh independence of Newton’s method for discretized variational problems. SIAM J. Control Optim. 39, 961-980.
• DONTCHEV, A.L. and MALANOWSKI, K. (2000) A characterization of Lipschitzian stability in optimal control. Calculus of Variations and Optimal Control (Haifa, 1998), 62-76, Chapman & Hall/CRC Res. Notes Math. 411 Chapman & Hall/CRC, Boca Raton, FL.
• DONTCHEV, A.L. and ROCKAFELLAR, R.T. (1996) Characterizations of strong regularity for variational inequalities over polyhedral convex sets, SIAM J. Optim. 6, 1087-1105.
• DONTCHEV, A.L. and ROCKAFELLAR, R.T. (2009) Implicit Functions and Solution Mappings. Springer Mathematics Monographs, Springer, Dordrecht.
• FELGENHAUER, U. (2008) The shooting approach in analyzing bang-bang extremals with simultaneous control switches. Control & Cybernetics 37, 307-327.
• FELGENHAUER, U., POGGIOLINI, L. and STEFANI, G. (2009) Optimality and stability result for bang-bang optimal controls with simple and double switch behavior. Control and Cybernetics, in this issue.
• KELLEY, C.T. (2003), Solving Nonlinear Equations with Newton’s Method. Fundamentals of Algorithms, SIAM, Philadelphia, PA.
• ROBINSON, S.M. (1980) Strongly regular generalized equations. Math. Oper. Res. 5, 43-62.
• ROBINSON, S.M. (1994) Newton’s method for a class of nonsmooth functions. Set-Valued Anal. 2, 291-305.