Abstract
The paper indicates which kind of approximate multifunctions allow to solve an inclusion (generalized equation) by a Newton-type method. It will be shown how the imposed condition can be applied to approximations of smooth and nonsmooth functions as well as of subdifferentials.
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References
F.H. Clarke, Optimization and Nonsmooth Analysis ( Wiley, New York, 1983 ).
B. Kummer, “Newton’s method for non-differentiable functions”, In: J. Guddat et al. eds, Advances in Math. Optimization (Akademie-Verlag Berlin, Ser. Mathem. Res. 45, 1988 ) 114–125.
B. Kummer, “On Stability and Newton-type methods for Lipschitzian equations with applications to optimization problems”, In: Lecture Notes in Control and Inform. Sci. 180, System Modelling and Optimization, P. Kali, ed., (Proceedings of the 15th IFIP Conference, Zürich 1991, Springer-Verlag, 1992 ) 3–16.
B. Kummer, “Newton’s method based on generalized derivatives for non-smooth functions: Convergence Analysis”, In: Lecture Notes in Econom. and Math. Systems 382, Advances in Optimization, W. Oettli, D. Pallaschke, eds., Proceedings 6th French-German Colloquium on Optimization, Lambrecht, FRG, 1991, ( Springer-Verlag, Berlin 1992 ) 171–194.
B.S. Mordukhovich, Approximation Methods in Problems of Optimization and (Theory of) Control, (in Russian) ( Nauka, Moskva, 1988 ).
J.-S. Pang, “Newton’s method for B-differentiable equations”, Math. of Oper. Res. 15 (1990) 311–341.
J.-S. Pang, S.A. Gabriel, “NE/SQP: A robust algorithm for the nonlinear complementarity problem”, Working Paper 1991, Department of Math. Sc., The Johns Hopkins Univ., Baltimore Maryland 21218.
J.-S. Pang, Liqun Qi, “Nonsmooth equations: motivation and algorithms”, Appl. Mathem. Preprint AM 91/22, The University of New South Wales, Kensington, New South Wales 2033, Australia.
K. Park, “Continuation methods for nonlinear programming”, Ph.D. Disseration, Department of Industrial Engineering, Univ. of Wisconsin-Madison 1989.
L. Qi, “Convergence analysis of some algorithms for solving nonsmooth equations”, Manuscript, 1991, School of Math., The Univ. of New South Wales, Kensington, New South Wales.
D. Ralph, “Global convergence of damped Newton’s method for nonsmooth equations, via the path search”, Techn. Report TR 90–1181 (1990), Department of Computer Sc., Cornell Univ. Ithaca, New York.
S.M. Robinson, “Newton’s method for a class of nonsmooth functions”, Working Paper, Aug. 1988, Univ. of Wisconsin-Madison, Department of Industrial Engineering, Madison, WI 53706.
R.T. Rockafellar, “Generalized directional derivatives and subgradients of nonconvex functions”, Canad. J. Math. 32 (1980) 257–280.
R.T. Rockafellar “Lipschitzian properties of multifunctions”, Nonlinear Anal. 9 (1985) 867–885.
L. Thibault, “Subdifferentials of compactly Lipschitzian vector-valued functions”, Ann. Mat. Pura Appl. (4) 125 (1980) 157–192.
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© 1995 Springer-Verlag Berlin Heidelberg
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Kummer, B. (1995). Approximation of Multifunctions and Superlinear Convergence. In: Durier, R., Michelot, C. (eds) Recent Developments in Optimization. Lecture Notes in Economics and Mathematical Systems, vol 429. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-46823-0_19
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DOI: https://doi.org/10.1007/978-3-642-46823-0_19
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