Local convergence of Newton’s method for solving generalized equations with monotone operator

References

• Dontchev AL . Local analysis of a Newton-type method based on partial linearization. In the mathematics of numerical analysis (Park City, UT, 1995). Providence (RI): American Mathematical Society; 1996.
   Google Scholar
• Dontchev AL . Local convergence of the Newton method for generalized equations. C R Acad Sci Paris Sér I Math. 1996;322(4):327–331.
   Google Scholar
• Dontchev AL , Rockafellar RT . Implicit functions and solution mappings. Springer monographs in mathematics. Dordrecht: Springer; 2009. A view from variational analysis.
   Google Scholar
• Dontchev AL , Rockafellar RT . Newton’s method for generalized equations: a sequential implicit function theorem. Math Program. 2010;123(1, Ser B):139–159.
   Web of Science ®Google Scholar
• Wang J . Convergence ball of Newton’s method for generalized equation and uniqueness of the solution. J Nonlinear Convex Anal. 2015;16(9):1847–1859.
   Web of Science ®Google Scholar
• Uko LU . Generalized equations and the generalized Newton method. Math Program. 1996;73(3, Ser A):251–268.
   Web of Science ®Google Scholar
• Josephy M . Newton’s method for generalized equations and the PIES energy model. Madison: University of Wisconsin-Madison; 1979.
   Google Scholar
• Ferreira OP , Silva GN . Inexact Newton’s method to nonlinear functions with values in a cone. 2015. arXiv:1510.01947.
   Google Scholar
• Robinson SM . Extension of Newton’s method to nonlinear functions with values in a cone. Numer Math. 1972;19:341–347.
   Web of Science ®Google Scholar
• Dokov SP , Dontchev AL . Robinson’s strong regularity implies robust local convergence of Newton’s method. Optimal control (Gainesville, FL, 1997). Vol. 15. Applied optimization. Dordrecht: Kluwer Academic Publishers; 1998. p. 116–129
   Google Scholar
• Wang Z . Extensions of Kantorovich theorem to complementarity problem. J Appl Math Mech. 2008;88(3):179–190.
   Google Scholar
• Pietrus A , Jean-Alexis C . Newton-secant method for functions with values in a cone. Serdica Math J. 2013;39(3–4):271–286.
   Google Scholar
• Daniel JW . Newton’s method for nonlinear inequalities. Numer Math. 1973;21:381–387.
   Web of Science ®Google Scholar
• Silva GN . Kantorovich’s theorem on Newton’s method for solving generalized equations under the majorant condition. Appl Math Comput. 2016;286:178–188.
   Web of Science ®Google Scholar
• Uko LU , Argyros IK . Generalized equations, variational inequalities and a weak Kantorovich theorem. Numer Algorithm. 2009;52(3):321–333.
   Web of Science ®Google Scholar
• Gonçalves MLN . Inexact Gauss-Newton like methods for injective-overdetermined systems of equations under a majorant condition. Numer Algorithm. 2016;72(2):377–392.
   Web of Science ®Google Scholar
• Ferreira OP , Gonçalves MLN , Oliveira PR . Local convergence analysis of inexact Gauss-Newton like methods under majorant condition. J Comput Appl Math. 2012;236(9):2487–2498.
   Web of Science ®Google Scholar
• Ferreira OP , Svaiter BF . Kantorovich’s majorants principle for Newton’s method. Comput Optim Appl. 2009;42(2):213–229.
   Web of Science ®Google Scholar
• Zabrejko PP , Nguen DF . The majorant method in the theory of Newton-Kantorovich approximations and the Pták error estimates. Numer Funct Anal Optim. 1987;9(5–6):671–684.
   Web of Science ®Google Scholar
• Wang X . Convergence of Newton’s method and inverse function theorem in Banach space. Math Comp. 1999;68(225):169–186.
   Web of Science ®Google Scholar
• Chang DC , Wang J , Yao JC . Newton’s method for variational inequality problems: Smale’s point estimate theory under the γ-condition. Appl Anal. 2015;94(1):44–55.
   Web of Science ®Google Scholar
• Smale S . Newton’s method estimates from data at one point. The merging of disciplines: new directions in pure, applied, and computational mathematics. New York (NY): Springer; 1986. p. 185–196
   Google Scholar
• Facchinei F , Pang JS . Finite-dimensional variational inequalities and complementary problems. New York (NY): Springer-Verlag; 2003.
   Google Scholar
• Ferreira O . Local convergence of Newton’s method in Banach space from the viewpoint of the majorant principle. IMA J Numer Anal. 2009;29(3):746–759.
   Web of Science ®Google Scholar
• Rall LB . A note on the convergence of Newton’s method. SIAM J Numer Anal. 1974;11(1):34–36.
   Web of Science ®Google Scholar
• Traub JF , Woźniakowski H . Convergence and complexity of Newton iteration for operator equations. J Assoc Comput Mach. 1979;26(2):250–258.
   Web of Science ®Google Scholar
• Blum L , Cucker F , Shub M , et al . Complexity and real computation. New York (NY): Springer-Verlag; 1998. With a foreword by Richard M. Karp.
   Google Scholar
• Ferreira OP , Gonçalves MLN , Oliveira PR . Local convergence analysis of the Gauss-Newton method under a majorant condition. J Complexity. 2011;27(1):111–125.
   Web of Science ®Google Scholar