Abstract
We present a Kantorovich-type semilocal convergence analysis of the Newton–Josephy method for solving a certain class of variational inequalities. By using a combination of Lipschitz and center-Lipschitz conditions, and our new idea of recurrent functions, we provide an analysis with the following advantages over the earlier works (Wang 2009, Wang and Shen, Appl Math Mech 25:1291–1297, 2004) (under the same or less computational cost): weaker sufficient convergence conditions, larger convergence domain, finer error bounds on the distances involved, and an at least as precise information on the location of the solution.
Similar content being viewed by others
References
Alefeld, G., Herzberger, J.: Introduction to Interval Computations. Translated from the German by Jon Rokne, Computer Science and Applied Mathematics. Academic, New York (1983)
Argyros, I.K.: On the Newton–Kantorovich hypothesis for solving equations. J. Comput. Appl. Math. 169, 315–332 (2004)
Argyros, I.K.: A unifying local–semilocal convergence analysis and applications for two-point Newton-like methods in Banach space. J. Math. Anal. Appl. 298, 374–397 (2004)
Argyros, I.K.: Convergence and Applications of Newton-Type Iterations. Springer, New York (2008)
Argyros, I.K., Hilout, S.: Efficient Methods for Solving Equations and Variational Inequalities. Polimetrica, Milan (2009)
Argyros, I.K., Hilout, S.: Enclosing roots of polynomial equations and their applications to iterative processes. Surveys Math. Appl. 4, 119–132 (2009)
Brézis, H.: Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert, North-Holland Mathematics Studies, No. 5, Notas de Matemática (50). North-Holland, Amsterdam (1973)
Chen, B., Harker, P.T.: A continuation method for monotone variational inequalities. Math. Program. 69, 273–253 (1995)
Eaves, B.C.: A Locally Quadratically Convergent Algorithm for Computing Stationary Point. Dpt of Operations Research, Stanford University, Palo Alto (1978)
Harker, P.T., Pang, J.S.: Finite-dimensional variational inequality and nonlinear complementarity problems: a survey of theory, algorithms and applications. Math. Program. Ser. B, 48, 161–220 (1990)
Izmailov, A.F, Solodov, M.V.: Inexact Josephy–Newton framework for generalized equations and its applications to local analysis of Newtonian methods for constrained optimization. Comput. Optim. Appl. doi:10.1007/s10589-009-9265-2
Josephy, N.H.: Newton’s method for generalized equations, Technical report no. 1965, Mathematics Research Center, University of Wisconsin, Madison, WI (1979)
Kantorovich, L.V., Akilov, G.P.: Functional Analysis. Pergamon, Oxford (1982)
Ortega, L.M., Rheinboldt, W.C.: Iterative Solution of Nonlinear Equations in Several Variables. Academic, New York (1970)
Peng, Ji-M., Kanzow, C., Fukushima, M.: A hybrid Josephy–Newton method for solving box constrained variational inequality problems via the D-gap function. Optim. Methods Softw. 10, 687–710 (1999)
Robinson, S.M.: Generalized equations and their solutions, part I: Basic theory. Math. Program. Stud. 10, 128–141 (1979)
Robinson, S.M.: Strongly regular generalized equations. Math. Oper. Res. 5, 43–62 (1980)
Solodov, M.V., Svaiter, B.F.: A new proximal-based globalization strategy for the Josephy–Newton method for variational inequalities. Optim. Methods Softw. 17, 965–983 (2002)
Wang, Z.Y.: Semilocal convergence of Newton’s method for finite-dimensional variational inequalities and nonlinear complementarity problems, Dissertation, Fakultät für Mathematik, Universität Karlsruhe (2005)
Wang, Z.Y.: Extensions of Kantorovich theorem to variational inequality problems. Z. Angew. Math. Mech. 88, 179–190 (2008)
Wang, Z.Y.: Extensions of the Newton–Kantorovich theorem to variational inequality problems. http://math.nju.edu.cn/~zywang/paper/Kantorovich_VIP.pdf (2009)
Wang, Z.Y., Shen, Z.H.: Kantorovich theorem for variational inequalities, (English Ed.). Appl. Math. Mech. 25, 1291–1297 (2004)
Xiao, B., Harker, P.T.: A nonsmooth Newton method for variational inequality. I: Theory, Math. Program. 65, 151–194 (1994)
Xiao, B., Harker, P.T.: A nonsmooth Newton method for variational inequality, II: numerical results. Math. Program. 65, 195–216 (1994)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Argyros, I.K., Hilout, S. A Kantorovich-type convergence analysis of the Newton–Josephy method for solving variational inequalities. Numer Algor 55, 447–466 (2010). https://doi.org/10.1007/s11075-010-9364-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11075-010-9364-2
Keywords
- Variational inequalities
- Newton–Josephy method
- Newton–Kantorovich hypothesis
- Majorizing sequence
- Fréchet-derivative