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A globally convergent LP-Newton method for piecewise smooth constrained equations: escaping nonstationary accumulation points

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Abstract

The LP-Newton method for constrained equations, introduced some years ago, has powerful properties of local superlinear convergence, covering both possibly nonisolated solutions and possibly nonsmooth equation mappings. A related globally convergent algorithm, based on the LP-Newton subproblems and linesearch for the equation’s infinity norm residual, has recently been developed. In the case of smooth equations, global convergence of this algorithm to B-stationary points of the residual over the constraint set has been shown, which is a natural result: nothing better should generally be expected in variational settings. However, for the piecewise smooth case only a property weaker than B-stationarity could be guaranteed. In this paper, we develop a procedure for piecewise smooth equations that avoids undesirable accumulation points, thus achieving the intended property of B-stationarity.

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Authors and Affiliations

  1. Faculty of Mathematics, Technische Universität Dresden, 01062, Dresden, Germany

    A. Fischer, M. Herrich & W. Scheck

  2. OR Department, VMK Faculty, Lomonosov Moscow State University, MSU, Uchebniy Korpus 2, Leninskiye Gory, Moscow, Russia, 119991

    A. F. Izmailov

  3. RUDN University, Miklukho-Maklaya Str. 6, Moscow, Russia, 117198

    A. F. Izmailov

  4. Department of Mathematic, Physics and Computer Sciences, Derzhavin Tambov State University, TSU, Internationalnaya 33, Tambov, Russia, 392000

    A. F. Izmailov

  5. IMPA – Instituto de Matemática Pura e Aplicada, Estrada Dona Castorina 110, Jardim Botânico, Rio de Janeiro, RJ, 22460-320, Brazil

    M. V. Solodov

Authors
  1. A. Fischer
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  2. M. Herrich
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  3. A. F. Izmailov
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  4. W. Scheck
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  5. M. V. Solodov
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Corresponding author

Correspondence to A. F. Izmailov.

Additional information

A. Fischer, M. Herrich, and W. Scheck were supported in part by the Volkswagen Foundation. A. F. Izmailov was supported by the Russian Science Foundation Grant 17-11-01168. M. V. Solodov was supported in part by CNPq Grants 303724/2015-3 and PVE 401119/2014-9, and by FAPERJ.

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Fischer, A., Herrich, M., Izmailov, A.F. et al. A globally convergent LP-Newton method for piecewise smooth constrained equations: escaping nonstationary accumulation points. Comput Optim Appl 69, 325–349 (2018). https://doi.org/10.1007/s10589-017-9950-5

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  • DOI: https://doi.org/10.1007/s10589-017-9950-5

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