Abstract
In this paper, we present the combination of the inexact Newton method and the generalized Newton method for solving nonsmooth equations F(x) = 0, characterizing the local convergence in terms of the perturbations and residuals. We assume that both iteration matrices taken from the B-differential and vectors F(x (k)) are perturbed at each step. Some results are motivated by the approach of Cătinaş regarding to smooth equations. We study the conditions, which determine admissible magnitude of perturbations to preserve the convergence of method. Finally, the utility of these results is considered based on some variant of the perturbed inexact generalized Newton method for solving some general optimization problems.
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Birgin, E.G., Krejić, N., Martínez, J.M.: Globally convergent inexact quasi-Newton methods for solving nonlinear systems. Numer. Algorithms 32, 249–260 (2003)
Bonettini, S.: A nonmonotone inexact Newton method. Optim. Methods Softw. 20, 475–491 (2005)
Bonettini, S., Tinti, F.: A nonmonotone semismooth inexact Newton method. Optim. Methods Softw. 22, 637–657 (2007)
Cătinaş, E.: Inexact perturbed Newton methods and application to a class of Krylov solvers. J. Optim. Theory Appl. 108, 543–571 (2001)
Cătinaş, E.: The inexact, inexact perturbed and quasi-Newton methods are equivalent models. Math. Comput. 74, 291–301 (2004)
Clarke, F.H.: Nonsmooth Analysis. Wiley, New York (1983)
Cores, D., Tapia, R.A.: Perturbation lemma for the Newton method with application to the SQP Newton method. J. Optim. Theory Appl. 97, 271-280 (1998)
Dembo, R.S., Eisenstat, S.C., Steihaug, T.: Inexact Newton methods. SIAM J. Numer. Anal. 14, 400–408 (1982)
Dennis, J.E., Schnabel, R.B.: Numerical Methods for Unconstrained Optimization and Nonlinear Equations. Prentice-Hall, Englewood Cliffs (1983)
Famularo, D., Sergeyev, Y.D., Pugliese, P.: Test problems for Lipschitz univariate global optimization with multiextremal constraints. In: Dzemyda, G., Saltenis, V., Žilinskas, A. (eds.) Stochastic and Global Optimization, pp. 93–109. Kluwer, Dordrecht (2002)
Floudas, C.A., Pardalos, P.M.: State of the Art in Global Optimization. Kluwer, Dordrecht (1996)
Gowda, M.S., Ravindran, G.: Algebraic univalence theorems for nonsmooth functions. J. Math. Anal. Appl. 252, 917–935 (2000)
Grapsa, T.N., Antonelou, G.E., Kostopoulos, A.E.: Perturbed Newton method for unconstrained optimization. In: Conference in Numerical Analysis NumAn 2007, Kalamata, Greece, 2007, pp. 77–80. http://www.math.upatras.gr/numan2007/NumAn2007.pdf (2007). Accessed 6 June 2012
Martínez, J.M.: Quasi-inexact-Newton methods with global convergence for solving constrained nonlinear systems. Nonlinear Anal. Theor. Meth. Appl. 30, 1–8 (1997)
Martínez, J.M., Qi, L.: Inexact Newton method for solving nonsmooth equations. J. Comput. Appl. Math. 60, 127–145 (1995)
Mifflin, R.: Semismooth and semiconvex functions in constrained optimization. SIAM J. Control Optim. 15, 959–972 (1977)
Ortega, J.M., Rheinboldt, W.C.: Iterative Solution of Nonlinear Equations in Several Variables. Academic Press, New York (1970)
Pang, J.S., Qi, L.: A globally convergent Newton method for convex SC1 minimization problem. J. Optim. Theory Appl. 85, 633–648 (1995)
Pardalos, P.M., Rosen, J.B. (eds.): Computational Methods in Global Optimization. In: Annals of Operation Research, Vol. 25. J.C. Baltzer AG, Basel (1990)
Patwardhan, A.A., Karim, M.N., Shah, R.: Controller tuning by a least-squares method. AIChE J. 33, 1735–1737 (1987)
di Pillo G.: Exact penalty method. In: Spedicato, E. (eds.) Algorithms for Continuous Optimization: The State of Art, pp. 209–253. Kluwer, Dordrecht (1994)
di Pillo G., Lucidi S.: On exact augmented Lagrangian functions in nonlinear programming problems. In: di Pillo, G., Gianessi, F. (eds.) Nonlinear Optimization and Applications, pp. 85–103. Plenum Press, New York (1996)
Pu, D., Tian, W.: Globally convergent inexact generalized Newton’s methods for nonsmooth equations. J. Comput. Appl. Math. 138, 37–49 (2002)
Qi, L., Chen, X.: A globally convergent successive approximation method for severely nonsmooth equations. SIAM J. Control Optim. 33, 402–418 (1995)
Qi, L., Sun, D.: A nonsmooth version of Newton’s method. Math. Program. 58, 353–367 (1993)
Qi, L.: Convergence analysis of some algorithms for solving nonsmooth equations. Math. Oper. Res. 18, 227–244 (1993)
Qi, L.: Superlinearly convergent approximate Newton methods for LC1 optimization problems. Math. Program. 64, 277–295 (1994)
Qi, L.: C-differential operators, C-differentiability and generalized Newton methods. Applied Mathematics Report AMR 96/5, University of New South Wales, Australia (1996)
Rademacher, H.: Ueber partielle und totale Differenzierbarkeit I. Math. Ann. 79, 340–359 (1919)
Ralston, P.A.S., Watson, K.R., Patwardhan, A.A., Deshpande, P.B.: A computer algorithm for optimized control. Ind. Eng. Chem. Prod. Res. Dev. 24, 1132–1136 (1985)
Schittkowski K.: More Test Examples for Nonlinear Programming Codes. Springer, Berlin (1987)
Sergeyev, Y.D., Daponte, P., Grimaldi, D., Molinaro, A.: Two methods for solving optimization problems arising in electronic measurements and electrical engineering. SIAM J. Optim. 10, 1–21 (1999)
Stewart, G.W.: Introduction to Matrix Computations. Academic Press, London (1973)
Śmietański, M.J.: Inexact quasi-Newton global convergent method for solving constrained nonsmooth equations. Int. J. Comput. Math. 84, 1157–1170 (2007)
Śmietański, M.J.: Some superlinearly convergent inexact quasi-Newton method for solving nonsmooth equations. Optim. Methods Softw. 27, 405–417 (2012)
Ypma, T.J.: The effect of rounding errors on Newton-like methods. IMA J. Numer. Anal. 3, 109–118 (1983)
Zhu, D.: Affine scaling inexact generalized Newton algorithm with interior backtracking technique for solving bound-constrained semismooth equations. J. Comput. Appl. Math. 187, 227–252 (2006)
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Śmietański, M.J. A perturbed version of an inexact generalized Newton method for solving nonsmooth equations. Numer Algor 63, 89–106 (2013). https://doi.org/10.1007/s11075-012-9613-7
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DOI: https://doi.org/10.1007/s11075-012-9613-7
Keywords
- Nonsmooth equations
- Inexact Newton method
- Inexact generalized Newton method
- Perturbation
- B-differential
- Superlinear convergence