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A note on Fejér-monotone sequences in product spaces and its applications to the dual convergence of augmented Lagrangian methods

Abstract

In a recent Math. Program. paper, Eckstein and Silva proposed a new error criterion for the approximate solutions of augmented Lagrangian subproblems. Based on a saddle-point formulation of the primal and dual problems, they proved that dual sequences generated by augmented Lagrangians under this error criterion are bounded and that their limit points are dual solutions. In this note, we prove a new result about the convergence of Fejér-monotone sequences in product spaces (which seems to be interesting by itself) and, as a consequence, we obtain the full convergence of the dual sequence generated by augmented Lagrangians under Eckstein and Silva’s criterion.

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Reference

  1. Eckstein, J., Silva, P.J.S.: A practical relative error criterion for augmented Lagrangians. Math. Program. 141(1–2, Ser. A), 319–348 (2013)

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Correspondence to M. Marques Alves.

Additional information

The work of M. Marques Alves was partially supported by CNPq Grants Nos. 305414/2011-9 and 406250/2013-8. The work of B. F. Svaiter was partially supported by CNPq Grants Nos. 474996/2013-1, 302962/2011-5, E-26/102.940/2011 CIENTISTA DO NOSSO ESTADO, and PRONEX-Optimization.

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Alves, M.M., Svaiter, B.F. A note on Fejér-monotone sequences in product spaces and its applications to the dual convergence of augmented Lagrangian methods. Math. Program. 155, 613–616 (2016). https://doi.org/10.1007/s10107-014-0849-y

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  • DOI: https://doi.org/10.1007/s10107-014-0849-y

Keywords

  • Augmented Lagrangians
  • Saddle-point problem
  • Error criterion
  • Fejér-monotone sequences

Mathematics Subject Classification

  • 90C25
  • 90C30