Journal of Optimization Theory and Applications

, Volume 159, Issue 3, pp 562–575 | Cite as

A Two-Phase Algorithm for a Variational Inequality Formulation of Equilibrium Problems

  • J. Y. Bello Cruz
  • P. S. M. Santos
  • S. Scheimberg


We introduce an explicit algorithm for solving nonsmooth equilibrium problems in finite-dimensional spaces. A particular iteration proceeds in two phases. In the first phase, an orthogonal projection onto the feasible set is replaced by projections onto suitable hyperplanes. In the second phase, a projected subgradient type iteration is replaced by a specific projection onto a halfspace. We prove, under suitable assumptions, convergence of the whole generated sequence to a solution of the problem. The proposed algorithm has a low computational cost per iteration and, some numerical results are reported.


Equilibrium problem Projection method Relaxed method 



The authors would like to extend their gratitude towards anonymous referees whose suggestions helped us to improve the presentation of this paper. First author was partially supported by Project CAPES-MES-CUBA 226/2012, PROCAD-nf-UFG/UnB/IMPA research and PRONEX-CNPq-FAPERJ.


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Copyright information

© Springer Science+Business Media New York 2012

Authors and Affiliations

  • J. Y. Bello Cruz
    • 1
  • P. S. M. Santos
    • 2
  • S. Scheimberg
    • 3
  1. 1.Universidade Federal de GoiásGoiâniaBrazil
  2. 2.Universidade Federal PiauíTeresinaBrazil
  3. 3.PESC/COPPE—Programa de Engenharia de Sistemas e Computação, IM-Instituto de MatemáticaUniversidade Federal do Rio de JaneiroRio de JaneiroBrazil

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