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Convergence of inexact Newton methods for generalized equations

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Abstract

For solving the generalized equation \(f(x)+F(x) \ni 0\), where \(f\) is a smooth function and \(F\) is a set-valued mapping acting between Banach spaces, we study the inexact Newton method described by

$$\begin{aligned} \left( f(x_k)+ D f(x_k)(x_{k+1}-x_k) + F(x_{k+1})\right) \cap R_k(x_k, x_{k+1}) \ne \emptyset , \end{aligned}$$

where \(Df\) is the derivative of \(f\) and the sequence of mappings \(R_k\) represents the inexactness. We show how regularity properties of the mappings \(f+F\) and \(R_k\) are able to guarantee that every sequence generated by the method is convergent either q-linearly, q-superlinearly, or q-quadratically, according to the particular assumptions. We also show there are circumstances in which at least one convergence sequence is sure to be generated. As a byproduct, we obtain convergence results about inexact Newton methods for solving equations, variational inequalities and nonlinear programming problems.

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Notes

  1. Actually, in his pioneering work [15] Robinson considered variational inequalities only.

  2. Since our analysis is local, one could localize these assumptions around a solution \(\bar{x}\) of (1). Also, in some of the presented results, in particular those involving strong metric subregularity, it is sufficient to assume continuity of \(Df\) only at \(\bar{x}\). Since the paper is already quite involved technically, we will not go into these refinements in order to simplify the presentation as much as possible.

  3. The classical inverse function theorem actually gives us more: it shows that the single-valued localization of the inverse is smooth and provides also the form of its derivative.

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Acknowledgments

The authors wish to thank the referees for their valuable comments on the original submission.

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Authors and Affiliations

  1. Mathematical Reviews, Ann Arbor, MI, 48107-8604, USA

    A. L. Dontchev

  2. Department of Mathematics, University of Washington, Seattle, WA, 98195-4350, USA

    R. T. Rockafellar

Authors
  1. A. L. Dontchev
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  2. R. T. Rockafellar
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Corresponding author

Correspondence to A. L. Dontchev.

Additional information

Dedicated to Jon Borwein on the occasion of his 60th birthday.

This work is supported by the National Science Foundation Grant DMS 1008341 through the University of Michigan.

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Dontchev, A.L., Rockafellar, R.T. Convergence of inexact Newton methods for generalized equations. Math. Program. 139, 115–137 (2013). https://doi.org/10.1007/s10107-013-0664-x

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  • DOI: https://doi.org/10.1007/s10107-013-0664-x

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