Abstract
We present a general iterative procedure for solving generalized equations in the nonsmooth framework. To this end, we consider a class of functions admitting a certain type of approximation and establish a local convergence theorem that one can apply to a wide range of particular problems.
Similar content being viewed by others
References
J.P. Aubin, “Lipschitz behavior of solutions to convex minimization problems,” Mathematics of Operations Research, vol. 9, pp. 87–111, 1984.
J.P. Aubin and H. Frankowska, Set-Valued Analysis, Birkhäuser, Boston, 1990.
A.L. Dontchev, “Local convergence of the Newton method for generalized equation,” C.R.A.S Paris, vol. 322, Serie I, pp. 327–331, 1996.
A.L. Dontchev, “Local analysis of a Newton-type method based on partial linearization,” in The Mathematics of Numerical Analysis, Renegar, James et al. (Eds.), 1995 AMS-SIAM Summer Seminar in Applied Mathematics, Providence, RI: AMS. Lect. Appl. Math., vol., 32, pp. 295–306, 1996.
A.L. Dontchev and W.W. Hager, “An inverse function theorem for set-valued maps,” Proc. Amer. Math. Soc., vol. 121, pp. 481–489, 1994.
A.L. Dontchev and R.T. Rockafellar, “Characterizations of strong regularity for variational inequalities over polyhedral convex sets,” SIAM J. Optim, vol. 6, no. 4, pp. 1087–1105, 1996.
A.L. Dontchev, “Uniform convergence of the Newton method for Aubin continuous maps,’ Serdica Math. J., vol. 22, no. 3, pp. 385–398, 1996.
F. Facchinei and J.-S. Pang, Finite-Dimensional Variational Inequalities and Complementarity Problems, Springer Series in Operations Research, Springer-Verlag, New York, 2003.
M.C. Ferris and J.S. Pang, “Engineering and economic applications of complementarity problems,” SIAM Rev., vol. 39, no. 4, pp. 669–713, 1997.
M.H. Geoffroy, S. Hilout, and A. Piétrus, “Acceleration of convergence in Donctchev’s iterative methods for solving variational inclusions,” Serdica Math. J., pp. 45–54, 2003.
M.H. Geoffroy and A. Piétrus, A Superquadratic Method for Solving Generalized Equations in the Hölder Case, Ricerche Di Matematica, vol. LII, fasc. 2, pp. 231–240, 2003.
M.H. Geoffroy and A. Piétrus, “Local convergence of some iterative methods for generalized equation,” J. Math. Anal. Appl., no. 2, pp. 497–505, 2004.
A.D. Ioffe and V.M. Tikhomirov, Theory of Extremal Problems, North Holland: Amsterdam, 1979.
B.S. Mordukhovich, “Complete characterization of openess metric regularity and Lipschitzian properties of multifunctions,” Trans. Amer. Math. Soc, vol. 340, pp. 1–36, 1993.
A. Pietrus, “Does Newton’s method converges uniformly in mild differentiability context?” Revista Colombiana de Matematicas, vol. 32, no. 2, pp. 49–56, 2000.
A. Pietrus, “Generalized equation under mild differentiability conditions,” Revista de la Real Academia de Ciencias Exactas de Madrid, vol. 94, no. 1, pp. 15–18, 2000.
S.M. Robinson, “Normal maps induced by linear transformations,” Math. Oper. Res., vol. 17, pp. 691–714, 1992.
S.M. Robinson, “Newton’s method for a class of nonsmooth functions,” Set-Valued Anal, vol. 2, pp. 291–305, 1994.
R.T. Rockafellar, “Lipschitzian properties of multifunctions, nonlinear analysis,” vol. 9, pp. 867–885, 1984.
R.T. Rockafellar and R. Wets, Variational Analysis. A Series of Comprehensive Studies in Mathematics, Springer, vol. 317, 1998.
Author information
Authors and Affiliations
Corresponding author
Additional information
Mathematics Subject Classification (2000): 47H04, 65K10
Rights and permissions
About this article
Cite this article
Geoffroy, M.H., Pietrus, A. A General Iterative Procedure for Solving Nonsmooth Generalized Equations. Comput Optim Applic 31, 57–67 (2005). https://doi.org/10.1007/s10589-005-1104-5
Received:
Revised:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/s10589-005-1104-5