Publication Cover
Applicable Analysis
An International Journal
Volume 98, 2019 - Issue 8
151
Views
5
CrossRef citations to date
0
Altmetric
Articles

Inexact Newton method for non-linear functions with values in a cone

Pages 1461-1477 | Received 15 Aug 2017, Accepted 13 Jan 2018, Published online: 29 Jan 2018

References

  • Blum L , Cucker F , Shub M , et al . Complexity and real computation. New York (NY): Springer-Verlag; 1998.
  • Dennis JE , Schnabel RB . Numerical methods for unconstrained optimization and nonlinear equations. Classics in applied mathematics. Society for industrial and applied mathematics (SIAM), Philadelphia (PA); 1996. Corrected reprint of the 1983 original.
  • Daniel JW . Newton’s method for nonlinear inequalities. Numer Math. 1973;21:381–387.
  • Deuflhard P . Newton methods for nonlinear problems. In: Springer series in computational mathematics. Vol. 35. Berlin: Springer-Verlag; 2004. Affine invariance and adaptive algorithms.
  • Dontchev AL , Rockafellar RT . Implicit functions and solution mappings. Springer monographs in mathematics. Dordrecht: Springer; 2009. A view from variational analysis.
  • He Y , Sun J . Error bounds for degenerate cone inclusion problems. Math Oper Res. 2005;30(3):701–717.
  • Krantz SG , Parks HR . The implicit function theorem. In: Modern Birkhäuser classics. New York (NY): Birkhäuser/Springer; 2013. History, theory, and applications, Reprint of the 2003 edition.
  • Li C , Ng KF . Convergence analysis of the Gauss–Newton method for convex inclusion and convex-composite optimization problems. J Math Anal Appl. 2012;389(1):469–485.
  • Pšeničnyĭ BN . A Newton method for the solution of systems of equalities and inequalities. Mat Zametki. 1970;8:635–640.
  • Robinson SM . Extension of Newton’s method to nonlinear functions with values in a cone. Numer Math. 1972;19:341–347.
  • Li C , Ng KF . Convergence analysis of the Gauss-Newton method for convex inclusion problems and convex composite optimization. Preprint. 2013;1–29.
  • Ferreira O . A robust semi-local convergence analysis of Newton’s method for cone inclusion problems in Banach spaces under affine invariant majorant condition. J Comput Appl Math. 2015;279:318–335.
  • Alvarez F , Bolte J , Munier J . A unifying local convergence result for Newton’s method in Riemannian manifolds. Found Comput Math. 2008;8(2):197–226.
  • Dedieu J-P , Priouret P , Malajovich G . Newton’s method on Riemannian manifolds: convariant alpha theory. IMA J Numer Anal. 2003;23(3):395–419.
  • Dembo RS , Eisenstat SC , Steihaug T . Inexact Newton methods. SIAM J Numer Anal. 1982;19(2):400–408.
  • Argyros IK , Hilout S . Inexact Newton-type methods. J. Complexity. 2010;26(6):577–590.
  • Dontchev AL , Rockafellar RT . Convergence of inexact Newton methods for generalized equations. Math Program. 2013;139(1–2 Ser B):115–137.
  • Ferreira OP , Svaiter BF . A robust Kantorovich’s theorem on the inexact Newton method with relative residual error tolerance. J Complexity. 2012;28(3):346–363.
  • Kelley C . Solving nonlinear equations with Newton’s method. Soc Ind Appl Math. 2003.
  • Adly S , Cibulka R , Ngai HV . Newton’s method for solving inclusions using set-valued approximations. SIAM J Optim. 2015;25(1):159–184.
  • Adly S , Van Ngai H , Nguyen VV . Newton’s method for solving generalized equations: Kantorovich’s and Smale’s approaches. J Math Anal Appl. 2016;439(1):396–418.
  • Cibulka R , Dontchev A , Preininger J , et al . Kantorovich-type theorems for generalized equations. Research Report. 2015–2016;2015:1–26.
  • Rashid MH . Convergence analysis of extended Hummel–Seebeck-type method for solving variational inclusions. Vietnam J Math. Dec 2016;44(4):709–726.
  • Bartle RG . Newton’s method in banach spaces. Proc Amer Math Soc. 1955;6(5):827–831.
  • Ciarlet PG , Mardare C . On the Newton–Kantorovich theorem. Anal Appl. 2012;10(3):249–269.
  • Dontchev AL . Local analysis of a Newton-type method based on partial linearization. In: The mathematics of numerical analysis. Vol. 32., Lectures in applied mathematics Providence (RI): American Mathematical Society; 1996. p. 295–306.
  • Ferreira OP , Svaiter BF . Kantorovich’s majorants principle for Newton’s method. Comput Optim Appl. 2009;42(2):213–229.
  • Wang X . Convergence of Newton’s method and inverse function theorem in Banach space. Math Comput. 1999;68(225):169–186.
  • Moret I . A Kantorovich-type theorem for inexact Newton methods. Numer Funct Anal Optim. 1989;10(3–4):351–365.
  • Shen W , Li C . Kantorovich-type convergence criterion for inexact Newton methods. Appl Numer Math. 2009;59(7):1599–1611.
  • Robinson SM . Normed convex processes. Trans Amer Math Soc. 1972;174:127–140.
  • Rockafellar RT . Monotone processes of convex and concave type. In: Memoirs of the American mathematical society. Vol. 77. Providence (RI): American Mathematical Society; 1967.
  • Rockafellar RT . Convex analysis. In: Princeton mathematical series. Vol. 28. Princeton (NJ): Princeton University Press; 1970.
  • Hiriart-Urruty J-B , Lemaréchal C . Convex analysis and minimization algorithms I. In: Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Vol. 305. Berlin: Springer-Verlag; 1993. Fundamentals.
  • Ferreira OP , Gonçalves MLN , Oliveira PR . Convergence of the Gauss-Newton method for convex composite optimization under a majorant condition. SIAM J Optim. 2013;23(3):1757–1783.
  • Cibulka R , Dontchev A , Geoffroy MH . Inexact Newton methods and Dennis–Moré theorems for nonsmooth generalized equations. SIAM J Control Optim. 2015;53(2):1003–1019.
  • Dontchev AL , Rockafellar RT . Newton’s method for generalized equations: a sequential implicit function theorem. Math Program. 2010;123(1 Ser B):139–159.
  • Pietrus A , Jean-Alexis C . Newton-secant method for functions with values in a cone. Serdica Math J. 2013;39(3–4):271–286.

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.