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Superlinear convergence of a Newton-type algorithm for monotone equations. (English) Zbl 1114.65055
Summary: We consider the problem of finding solutions of systems of monotone equations. The Newton-type algorithm proposed in [M. V. Solodov and B. F. Svaiter, in: Reformulation: nonsmooth, piecewise smooth, semismooth and smoothing methods (Lausanne, 1997), Boston: Kluwer, Appl. Optim. 22, 355–369 (1999; Zbl 0928.65059)] 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.

MSC:
65H10 Numerical computation of solutions to systems of equations
90C53 Methods of quasi-Newton type
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[8] Kanzow, C., Yamashita, N., and Fukushima, M., Levenberg-Marquardt Methods for Constrained Nonlinear Equations with Strong Local Convergence Properties, Technical Report 2002-007, Department of Applied Mathematics and Physics, Kyoto University, Kyoto, Japan, 2002. · Zbl 1064.65037
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[15] Mangasarian, O. L., A Newton Method for Linear Programming, Data Mining Institute, Technical Report 02-02, 2002.
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