A modified Newton-secant method for solving nonsmooth generalized equations
Abstract
In this paper, we study the solvability of nonsmooth generalized equations in Banach spaces using a modified Newton-secant method, by assuming a Hölder condition. Also, we generalize a Dennis-Moré theorem to characterize the superlinear convergence of the proposed method applied to nonsmooth generalized equations under strong metric subregularity. Numerical examples are provided to illustrate the effectiveness of our approach.
Keyword : Newton-Kantorovich theorem, divided differences, Newton-secant method, generalized equations
How to Cite
de Sousa Amaral, V., dos Santos, P. S. M., Silva, G. N., & Souza, S. (2024). A modified Newton-secant method for solving nonsmooth generalized equations. Mathematical Modelling and Analysis, 29(2), 347–366. https://doi.org/10.3846/mma.2024.18680
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
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L.M. Graves. Some mapping theorems. Duke Math. J., 17:111–114, 1950. https://doi.org/10.1215/S0012-7094-50-01713-3
M.A. Hernández and M.J. Rubio. Semilocal convergence of the secant method under mild convergence conditions of differentiability. Comp. Math. Appl., 44(34):277–285, 2002. https://doi.org/10.1016/S0898-1221(02)00147-5
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W.J. Leong, M.A. Hassan and M.W. Yusuf. A matrix-free quasi-Newton method for solving large-scale nonlinear systems. Comput. Math. Appl., 62(5):2354–2363, 2011. https://doi.org/10.1016/j.camwa.2011.07.023
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M.H. Rashid, J.H. Wang and C. Li. Convergence analysis of a method for variational inclusions. Appl. Anal., 91(10):1943–1956, 2012. https://doi.org/10.1080/00036811.2011.618127
J. Rokne. Newton’s method under mild differentiability conditions with error analysis. Numer. Math., 18(5):401–412, 1971. https://doi.org/10.1007/BF01406677
P.S.M. Santos, G.N. Silva and R.C.M. Silva. Newton-type method for solving generalized inclusion. Numer. Algor., 88(1):1811–1829, 2021. https://doi.org/10.1007/s11075-021-01096-8
S. Bernard, C. Cabuzel, S.P. Nuiro and A. Piétrus. Extended semismooth Newton method for functions with values in a cone. Acta Appl. Math., 155(1):85–98, 2018. https://doi.org/10.1007/s10440-017-0146-x
E. Catinas. On some iterative methods for solving nonlinear equations. Rev. Anal. Numér. Théor. Approx., 23(1):47–53, 1994.
R. Cibulka, A.L. Dontchev and M.H. Geoffroy. Inexact Newton methods and Dennis–Moré theorems for nonsmooth generalized equations. SIAM J. Control Optim., 53(2):1003–1019, 2015. https://doi.org/10.1137/140969476
F.R. de Oliveira, O.P. Ferreira and G.N. Silva. Newton’s method with feasible inexact projections for solving constrained generalized equations. Comput. Optim. Appl., 72(1):159–177, 2019. https://doi.org/10.1007/s10589-018-0040-0
J.E. Dennis and J.J. Moré. A characterization of superlinear convergence and its application to quasi-Newton methods. Math. Comput., 28(126):549–560, 1974. https://doi.org/10.1090/S0025-5718-1974-0343581-1
E.D. Dolan and J.J. Moré. Benchmarking optimization software with performance profiles. Math. Program., 91(16):201–213, 2002. https://doi.org/10.1007/s101070100263
A.L. Dontchev. The Graves theorem revisited. J. Convex Anal., 3:45–53, 1996.
A.L. Dontchev. Local analysis of a Newton-type method based on partial linearization. Lect. Appl. Math., 32(1):295–306, 1996.
A.L. Dontchev. Uniform convergence of the Newton method for Aubin continuous maps. Serdica Math. J., 22(3):385–398, 1996. Available from Internet: http://eudml.org/doc/11643
A.L. Dontchev. Generalizations of the Dennis–Moré theorem. SIAM J. Optim., 22(3):821–830, 2012. https://doi.org/10.1137/110833567
A.L. Dontchev and W.W. Hager. An inverse mapping theorem for set-valued maps. Proc. Amer. Math. Soc., 121(2):481–489, 1994. https://doi.org/10.1090/S0002-9939-1994-1215027-7
A.L. Dontchev and R.T. Rockafellar. Characterizations of strong regularity for variational inequalities over polyhedral convex sets. SIAM J. Optim., 6(4):1087– 1105, 1996. https://doi.org/10.1137/S1052623495284029
A.L. Dontchev and R.T. Rockafellar. Implicit functions and solution mappings: A view from variational analysis, volume 616. Springer, 2009. https://doi.org/10.1007/978-1-4939-1037-3
O.P. Ferreira and G.N. Silva. Local convergence analysis of Newton’s method for solving strongly regular generalized equations. J. Math. Anal. Appl., 458(1):481– 496, 2018. https://doi.org/10.1016/j.jmaa.2017.09.023
M.H. Geoffroy and A. Piétrus. Local convergence of some iterative methods for generalized equations. J. Math. Anal. Appl., 290(2):497–505, 2004. https://doi.org/10.1016/j.jmaa.2003.10.008
L.M. Graves. Some mapping theorems. Duke Math. J., 17:111–114, 1950. https://doi.org/10.1215/S0012-7094-50-01713-3
M.A. Hernández and M.J. Rubio. Semilocal convergence of the secant method under mild convergence conditions of differentiability. Comp. Math. Appl., 44(34):277–285, 2002. https://doi.org/10.1016/S0898-1221(02)00147-5
C. Jean-Alexis and A. Piétrus. On the convergence of some methods for variational inclusions. Rev. R. Acad. Cien. serie A. Mat., 102(2):355–361, 2008. https://doi.org/10.1007/BF03191828
W.J. Leong, M.A. Hassan and M.W. Yusuf. A matrix-free quasi-Newton method for solving large-scale nonlinear systems. Comput. Math. Appl., 62(5):2354–2363, 2011. https://doi.org/10.1016/j.camwa.2011.07.023
Q. Liu and J. Wang. A one-layer recurrent neural network for constrained nonsmooth optimization. IEEE Trans. Syst. Man Cybern. Syst. Cybernetics, 41(5):1323–1333, 2011. https://doi.org/10.1109/TSMCB.2011.2140395
M.H. Rashid, J.H. Wang and C. Li. Convergence analysis of a method for variational inclusions. Appl. Anal., 91(10):1943–1956, 2012. https://doi.org/10.1080/00036811.2011.618127
J. Rokne. Newton’s method under mild differentiability conditions with error analysis. Numer. Math., 18(5):401–412, 1971. https://doi.org/10.1007/BF01406677
P.S.M. Santos, G.N. Silva and R.C.M. Silva. Newton-type method for solving generalized inclusion. Numer. Algor., 88(1):1811–1829, 2021. https://doi.org/10.1007/s11075-021-01096-8