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On the Application of the SCD Semismooth* Newton Method to Variational Inequalities of the Second Kind

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Abstract

The paper starts with a description of SCD (subspace containing derivative) mappings and the SCD Newton method for the solution of general inclusions. This method is then applied to a class of variational inequalities of the second kind. As a result, one obtains an implementable algorithm which exhibits locally superlinear convergence. Thereafter we suggest several globally convergent hybrid algorithms in which one combines the SCD Newton method with selected splitting algorithms for the solution of monotone variational inequalities. Finally, we demonstrate the efficiency of one of these methods via a Cournot-Nash equilibrium, modeled as a variational inequality of the second kind, where one admits really large numbers of players (firms) and produced commodities.

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Acknowledgements

The authors are deeply indebted to the reviewers for careful reading and important suggestions and remarks.

Funding

The research of the first author was supported by the Austrian Science Fund (FWF) under grant P29190-N32. The research of the second author was supported by the Grant Agency of the Czech Republic, Project 21-06569K, and the Australian Research Council, Project DP160100854. The research of the third author was supported by the Grant Agency of the Czech Republic, Project 21-06569K.

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Authors and Affiliations

  1. Institute of Computational Mathematics, Johannes Kepler University Linz, A-4040, Linz, Austria

    Helmut Gfrerer

  2. Institute of Information Theory and Automation, Czech Academy of Sciences, 18208, Prague, Czech Republic

    Jiří V. Outrata & Jan Valdman

  3. Centre for Informatics and Applied Optimization, Federation University of Australia, POB 663,, Ballarat, Vic, 3350, Australia

    Jiří V. Outrata

  4. Department of Applied Mathematics, Faculty of Information Technology, Czech Technical University in Prague, Thákurova 9, 16000, Prague 6, Czech Republic

    Jan Valdman

Authors
  1. Helmut Gfrerer
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  2. Jiří V. Outrata
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  3. Jan Valdman
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Correspondence to Helmut Gfrerer.

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Gfrerer, H., Outrata, J.V. & Valdman, J. On the Application of the SCD Semismooth* Newton Method to Variational Inequalities of the Second Kind. Set-Valued Var. Anal 30, 1453–1484 (2022). https://doi.org/10.1007/s11228-022-00651-2

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  • DOI: https://doi.org/10.1007/s11228-022-00651-2

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