Abstract
This paper presents sufficient and necessary conditions for the convergence of Newton’s method based on generalized derivatives. These conditions require uniform injectivity of the derivatives as well as uniform high-order approximation of the original locally Lipschitz function along rays through the solution. Our approach permits to determine approximate solutions of the Newton subproblems and to use such concepts of derivatives for nonsmooth functions, multivalued or not, as directional and B-derivatives, contingent derivatives, generalized Jacobians and others. Furthermore, we ensure solvability of the subproblems via surjecivi-ty of the derivatives and verify a Kantorovich-type convergence theorem.
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References
Aubin, J.P. Differential Calculus of set-valued maps. an update. Working Paper WP-87–93, (1987) IIASA, Laxenburg Austria
Aubin, J.P. & Ekeland, I. Applied Nonlinear Analysis. Wiley, New York, 1984
Aubin, J.P. & Frankowska, H. Set-valued Analysis. Birkhaüser, Basel, 1990
Aze, D. An inversion theorem for set-valued maps. Bull. Austral. Math. Soc. 37 (1988) pp. 411–414
Clarke, F.H. On the inverse function theorem. Pacific Journ. Math. 64, No. 1 (1976) pp. 97–102
Clarke, F.H. Optimization and Nonsmooth Analysis. Wiley, New York, 1983
Ekeland, I. On the variational principle. J. Math. Analysis & Appl. 47, No.2 (1974) pp. 324–353
Haraux, A. How to differentiate the projection on a convex set in Hilbert space. Some applications to variational inequalities. Journ. Math. Soc. Japan 29, (1977) pp. 615–631
Harker, P.T. & Pang, J.-S. Finite-dimensional variational inequality and nonlinear complementarity problems: A survey of theory, algorithms and applications. Mathematical Programming 48, (1990) pp. 161–220
Ioffe, A.D. Nonsmooth analysis: Differential calculus of nondifferentiable mappings. Trans. Amer. Math. Soc. 266, (1981) pp. 1–56
Ioffe, A.D. Global surjection and global inverse mapping theorems in Banach spaces. Ann. NewYork Acad.Sci. 491, (1987) pp. 181–189
Jittorntrum, K. Solution point differentiability without strict complementarity in nonlinear programming. Math. Programming Study 21, (1984) pp. 127–138
Jongen, H.Th., Klatte, D., Tammer, K. Implicit functions and sensitivity of stationary points. Mathematical Programming 49 (1990) pp. 123–138
Kojima, M. Strongly stable stationary solutions in nonlinear programs. In: Analysis and Computation of Fixed Points, S.M. Robinson ed., Academic Press, New York, 1980 pp. 93–138
Kojima, M. & Shindo, S. Extensions of Newton and quasi-Newton methods to systems of PC equations. Journ. of Operations Research Soc. of Japan 29 (1987) pp. 352–374
Kummer, B. Newton’s method for non-differentiable functions. In: Advances in Math. Optimization, J.Guddat et al. eds. Akademie Verlag Berlin, Ser. Mathem. Res. Vol. 45, 1988 pp. 114–125
Kummer, B. An implicit function theorem for C 0,1 -equations and parametric C 1,1 -optimization. Journ. Math. Analysis & Appl. (1991) Vol. 158, No.1, pp. 35–46
Kummer, B. Lipschitzian inverse functions, directional derivatives and application in C 1,1 -optimization. Working Paper WP- 89–084 (1989) IIASA Laxenburg, Austria
Langenbach, A. Über lipschitzstetige implizite Funktionen. Zeitschr.für Analysis und ihre Anwendungen Bd. 8 (3), (1989) pp. 289–292
Langenbach, A. Bemerkungen zur Theorie der lokal bzw. global invertierbaren R m -Abbildungen. Math.Nachr. 148, (1990) pp. 59–63
Ortega J.M. & Rheinboldt W.C. Iterative Solution of Nonlinear Equations of Several Variables. Academic Press, San Diego, 1970
Pang, J.-S. Newton’s method for B-differentiable equations. Math. of Operations Res. 15, (1990) pp. 311–341
Pang, J.-S. & Gabriel, S.A. NE/SQP: A robust algorithm for the nonlinear complementarity problem. Working Paper, (1991), Department of Math. Sc, The Johns Hopkins Univ., Baltimore Maryland 21218
Park, K. Continuation methods for nonlinear programming. Ph.D. Dissertation, (1989), Department of Industrial Engineering, Univ. of Wisconsin-Madison
Qi, L. Convergence analysis of some algorithms for solving nonsmooth equations. Manuscript, (1991), School of Math., The Univ. of New South Wales, Kensington, New South Wales
Ralph, D. 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
Robinson, S.M. Newton’s method for a class of nonsmooth functions. Working Paper, (1988), Univ. of Wisconsin-Madison, Department of Industrial Engineering, Madison, WI 53706
Robinson, S.M. An implicit-function theorem for B-differen-tiable functions. Working Paper WP-88–67, (1988), IIASA Laxenburg, Austria
Robinson, S.M. An implicit-function theorem for a class of nonsmooth functions. Mathematics of OR, Vol. 16, No. 2, May (1991), pp. 292–309
Thibault, L. Subdifferentials of compactly Lipschitzian vector-valued functions. Ann. Mat. Pura Appl. (4) 125, (1980) pp. 157–192
Thibault, L. On generalized differentials and subdifferentials of Lipschitz vector-valued functions. Nonlinear Analysis Theory Methods Appl. 6 (10), (1982) pp. 1037–1053
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Kummer, B. (1992). Newton’s Method Based on Generalized Derivatives for Nonsmooth Functions: Convergence Analysis. In: Oettli, W., Pallaschke, D. (eds) Advances in Optimization. Lecture Notes in Economics and Mathematical Systems, vol 382. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-51682-5_12
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DOI: https://doi.org/10.1007/978-3-642-51682-5_12
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