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Superlinear Convergence of a Newton-Type Algorithm for Monotone Equations

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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.

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Authors and Affiliations

  1. Research Associate, Department of Mathematics and Statistics, Curtin University of Technology, Perth, Western Australia, Australia

    G. Zhou

  2. Associate Professor, Department of Mathematics, National University of Singapore, Kent Ridge, Singapore, Republic of Singapore

    K. C. Toh

Authors
  1. G. Zhou
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  2. K. C. Toh
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Additional information

This research was supported by the Singapore-MIT Alliance and the Australian Research Council.

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Zhou, G., Toh, K.C. Superlinear Convergence of a Newton-Type Algorithm for Monotone Equations. J Optim Theory Appl 125, 205–221 (2005). https://doi.org/10.1007/s10957-004-1721-7

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  • DOI: https://doi.org/10.1007/s10957-004-1721-7

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