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Inexact Josephy–Newton framework for generalized equations and its applications to local analysis of Newtonian methods for constrained optimization

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Abstract

We propose and analyze a perturbed version of the classical Josephy–Newton method for solving generalized equations. This perturbed framework is convenient to treat in a unified way standard sequential quadratic programming, its stabilized version, sequential quadratically constrained quadratic programming, and linearly constrained Lagrangian methods. For the linearly constrained Lagrangian methods, in particular, we obtain superlinear convergence under the second-order sufficient optimality condition and the strict Mangasarian–Fromovitz constraint qualification, while previous results in the literature assume (in addition to second-order sufficiency) the stronger linear independence constraint qualification as well as the strict complementarity condition. For the sequential quadratically constrained quadratic programming methods, we prove primal-dual superlinear/quadratic convergence under the same assumptions as above, which also gives a new result.

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Authors and Affiliations

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

    A. F. Izmailov

  2. 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. 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 08-01-90001-Bel, and by RF President’s Grant NS-693.2008.1 for the support of leading scientific schools. The second author is supported in part by CNPq Grants 300513/2008-9 and 471267/2007-4, by PRONEX–Optimization, and by FAPERJ.

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Izmailov, A.F., Solodov, M.V. Inexact Josephy–Newton framework for generalized equations and its applications to local analysis of Newtonian methods for constrained optimization. Comput Optim Appl 46, 347–368 (2010). https://doi.org/10.1007/s10589-009-9265-2

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