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Examples of dual behaviour of Newton-type methods on optimization problems with degenerate constraints

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Abstract

We discuss possible scenarios of behaviour of the dual part of sequences generated by primal-dual Newton-type methods when applied to optimization problems with nonunique multipliers associated to a solution. Those scenarios are: (a) failure of convergence of the dual sequence; (b) convergence to a so-called critical multiplier (which, in particular, violates some second-order sufficient conditions for optimality), the latter appearing to be a typical scenario when critical multipliers exist; (c) convergence to a noncritical multiplier. The case of mathematical programs with complementarity constraints is also discussed. We illustrate those scenarios with examples, and discuss consequences for the speed of convergence. We also put together a collection of examples of optimization problems with constraints violating some standard constraint qualifications, intended for preliminary testing of existing algorithms on degenerate problems, or for developing special new algorithms designed to deal with constraints degeneracy.

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Authors and Affiliations

  1. Faculty of Computational Mathematics and Cybernetics, Department of Operations Research, Moscow State University, Leninskiye Gori, GSP-2, 119992, Moscow, Russia

    A. F. Izmailov

  2. 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. F. Izmailov
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  2. M. V. Solodov
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Correspondence to M. V. Solodov.

Additional information

Research of the first author is supported by the Russian Foundation for Basic Research Grants 07-01-00270, 07-01-00416 and 07-01-90102-Mong, and by RF President’s Grant NS-9344.2006.1 for the support of leading scientific schools. The second author is supported in part by CNPq Grants 301508/2005-4, 490200/2005-2 and 550317/2005-8, by PRONEX–Optimization, and by FAPERJ Grant E-26/151.942/2004.

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Izmailov, A.F., Solodov, M.V. Examples of dual behaviour of Newton-type methods on optimization problems with degenerate constraints. Comput Optim Appl 42, 231–264 (2009). https://doi.org/10.1007/s10589-007-9074-4

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  • DOI: https://doi.org/10.1007/s10589-007-9074-4

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