A relaxed constant positive linear dependence constraint qualification and applications

Abstract

In this work we introduce a relaxed version of the constant positive linear dependence constraint qualification (CPLD) that we call RCPLD. This development is inspired by a recent generalization of the constant rank constraint qualification by Minchenko and Stakhovski that was called RCRCQ. We show that RCPLD is enough to ensure the convergence of an augmented Lagrangian algorithm and that it asserts the validity of an error bound. We also provide proofs and counter-examples that show the relations of RCRCQ and RCPLD with other known constraint qualifications. In particular, RCPLD is strictly weaker than CPLD and RCRCQ, while still stronger than Abadie’s constraint qualification. We also verify that the second order necessary optimality condition holds under RCRCQ.

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Correspondence to Paulo J. S. Silva.

Additional information

This work was supported by PRONEX-Optimization (PRONEX-CNPq/FAPERJ E-26/171.510/2006-APQ1), Fapesp (Grants 2006/53768-0 and 2009/09414-7) and CNPq (Grants 300900/2009-0, 303030/2007-0 and 474138/2008-9).

The second author would like to acknowledge that he was a research assistant at the University of São Paulo while this work was carried out.

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Andreani, R., Haeser, G., Schuverdt, M.L. et al. A relaxed constant positive linear dependence constraint qualification and applications. Math. Program. 135, 255–273 (2012). https://doi.org/10.1007/s10107-011-0456-0

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Keywords

  • Nonlinear programming
  • Constraint qualifications
  • Augmented Lagrangian
  • Error bound property

Mathematics Subject Classification (2000)

  • 90C30
  • 49K99
  • 65K05