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Superlinear Convergence of a Newton-Type Algorithm for Monotone Equations

Author

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  • G. Zhou

    (Curtin University of Technology)

  • K. C. Toh

    (National University of Singapore)

Abstract

We consider the problem of finding solutions of systems of monotone equations. The Newton-type algorithm proposed in Ref. 1 has a very nice global convergence property in that the whole sequence of iterates generated by this algorithm converges to a solution, if it exists. Superlinear convergence of this algorithm is obtained under a standard nonsingularity assumption. The nonsingularity condition implies that the problem has a unique solution; thus, for a problem with more than one solution, such a nonsingularity condition cannot hold. In this paper, we show that the superlinear convergence of this algorithm still holds under a local error-bound assumption that is weaker than the standard nonsingularity condition. The local error-bound condition may hold even for problems with nonunique solutions. As an application, we obtain a Newton algorithm with very nice global and superlinear convergence for the minimum norm solution of linear programs.

Suggested Citation

  • G. Zhou & K. C. Toh, 2005. "Superlinear Convergence of a Newton-Type Algorithm for Monotone Equations," Journal of Optimization Theory and Applications, Springer, vol. 125(1), pages 205-221, April.
    • Handle: RePEc:spr:joptap:v:125:y:2005:i:1:d:10.1007_s10957-004-1721-7
    DOI: 10.1007/s10957-004-1721-7
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    References listed on IDEAS

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    1. C. Kanzow & H. Qi & L. Qi, 2003. "On the Minimum Norm Solution of Linear Programs," Journal of Optimization Theory and Applications, Springer, vol. 116(2), pages 333-345, February.
    2. ,, 2002. "Problems And Solutions," Econometric Theory, Cambridge University Press, vol. 18(4), pages 1007-1017, August.
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    5. ,, 2002. "Problems And Solutions," Econometric Theory, Cambridge University Press, vol. 18(5), pages 1273-1289, October.
    6. Hiroshige Dan & Nobuo Yamashita & Masao Fukushima, 2002. "A Superlinearly Convergent Algorithm for the Monotone Nonlinear Complementarity Problem Without Uniqueness and Nondegeneracy Conditions," Mathematics of Operations Research, INFORMS, vol. 27(4), pages 743-753, November.
    7. ,, 2002. "Problems And Solutions," Econometric Theory, Cambridge University Press, vol. 18(3), pages 819-821, June.
    8. ,, 2002. "Problems And Solutions," Econometric Theory, Cambridge University Press, vol. 18(2), pages 541-545, April.
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    Cited by:

    1. Chuanwei Wang & Yiju Wang & Chuanliang Xu, 2007. "A projection method for a system of nonlinear monotone equations with convex constraints," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 66(1), pages 33-46, August.
    2. Najib Ullah & Abdullah Shah & Jamilu Sabi’u & Xiangmin Jiao & Aliyu Muhammed Awwal & Nuttapol Pakkaranang & Said Karim Shah & Bancha Panyanak, 2023. "A One-Parameter Memoryless DFP Algorithm for Solving System of Monotone Nonlinear Equations with Application in Image Processing," Mathematics, MDPI, vol. 11(5), pages 1-26, March.
    3. Awwal, Aliyu Muhammed & Kumam, Poom & Abubakar, Auwal Bala, 2019. "Spectral modified Polak–Ribiére–Polyak projection conjugate gradient method for solving monotone systems of nonlinear equations," Applied Mathematics and Computation, Elsevier, vol. 362(C), pages 1-1.
    4. Eltiyeb Ali & Salem Mahdi, 2023. "Adaptive Hybrid Mixed Two-Point Step Size Gradient Algorithm for Solving Non-Linear Systems," Mathematics, MDPI, vol. 11(9), pages 1-35, April.
    5. Papp, Zoltan & Rapajić, Sanja, 2015. "FR type methods for systems of large-scale nonlinear monotone equations," Applied Mathematics and Computation, Elsevier, vol. 269(C), pages 816-823.

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