Journal of Optimization Theory and Applications

, Volume 168, Issue 1, pp 198–215 | Cite as

A Variant of the Hybrid Proximal Extragradient Method for Solving Strongly Monotone Inclusions and its Complexity Analysis

  • Maicon Marques Alves
  • B. F. Svaiter


This paper presents and studies the iteration-complexity of a variant of the hybrid proximal extragradient method for solving inclusion problems with strongly (maximal) monotone operators. As applications, we propose and analyze two special cases: variants of the Tseng’s forward–backward method for solving monotone inclusions with strongly monotone and Lipschitz continuous operators and of the Korpelevich extragradient method for solving (strongly monotone) variational inequalities.


Hybrid proximal extragradient method Strongly monotone operators Variational inequalities Tseng’s forward–backward method Korpelevich extragradient method 

Mathematics Subject Classification

47H05 47J20 90C060 90C33 65K10 



M. Marques Alves was partially supported by CNPq Grant Nos. 406250/2013-8, 237068/2013-3 and 306317/2014-1. B. F. Svaiter was partially supported by CNPq Grant Nos. 474996/2013-1, 302962/2011-5 and FAPERJ Grant E-26/201.584/2014.


  1. 1.
    Martinet, B.: Régularisation d’inéquations variationnelles par approximations successives. Rev. Française Inform. Rech. Opérationnelle 4(Ser. R–3), 154–158 (1970)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Rockafellar, R.T.: Monotone operators and the proximal point algorithm. SIAM J. Control Optim. 14(5), 877–898 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Solodov, M.V., Svaiter, B.F.: A hybrid approximate extragradient-proximal point algorithm using the enlargement of a maximal monotone operator. Set-Valued Anal. 7(4), 323–345 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Solodov, M.V., Svaiter, B.F.: A hybrid projection-proximal point algorithm. J. Convex Anal. 6(1), 59–70 (1999)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Burachik, R.S., Iusem, A., Svaiter, B.F.: Enlargement of monotone operators with applications to variational inequalities. Set-Valued Anal. 5(2), 159–180 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Monteiro, R.D.C., Svaiter, B.F.: On the complexity of the hybrid proximal extragradient method for the iterates and the ergodic mean. SIAM J. Optim. 20(6), 2755–2787 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Monteiro, R.D.C., Ortiz, C., Svaiter, B.F.: A first-order block-decomposition method for solving two-easy-block structured semidefinite programs. Math. Program. Comput. 6(2), 103–150 (2014)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Monteiro, R.D.C., Ortiz, C., Svaiter, B.F.: Implementation of a block-decomposition algorithm for solving large-scale conic semidefinite programming problems. Comput. Optim. Appl. 57(1), 45–69 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Monteiro, R.D.C., Svaiter, B.F.: Complexity of variants of Tseng’s modified F-B splitting and Korpelevich’s methods for hemivariational inequalities with applications to saddle-point and convex optimization problems. SIAM J. Optim. 21(4), 1688–1720 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Monteiro, R.D.C., Svaiter, B.F.: Iteration-complexity of a Newton proximal extragradient method for monotone variational inequalities and inclusion problems. SIAM J. Optim. 22(3), 914–935 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Monteiro, R.D.C., Svaiter, B.F.: An accelerated hybrid proximal extragradient method for convex optimization and its implications to second-order methods. SIAM J. Optim. 23(2), 1092–1125 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Monteiro, R.D.C., Svaiter, B.F.: Iteration-complexity of block-decomposition algorithms and the alternating direction method of multipliers. SIAM J. Optim. 23(1), 475–507 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Solodov, M.V., Svaiter, B.F.: Forcing strong convergence of proximal point iterations in a Hilbert space. Math. Program. 87(1, Ser. A), 189–202 (2000)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Solodov, M.V., Svaiter, B.F.: A truly globally convergent Newton-type method for the monotone nonlinear complementarity problem. SIAM J. Optim. 10(2), 605–625 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Tikhonov, A.N., Arsenin, V.Y.: Solutions of Ill-Posed Problems. Winston and Sons, Washington (1977)zbMATHGoogle Scholar
  16. 16.
    Nesterov, Y., Scrimali, L.: Solving strongly monotone variational and quasi-variational inequalities. Discret Contin. Dyn. Syst. 31(4), 1383–1396 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Brøndsted, A., Rockafellar, R.T.: On the subdifferentiability of convex functions. Proc. Am. Math. Soc. 16, 605–611 (1965)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Departamento de MatemáticaUniversidade Federal de Santa CatarinaFlorianópolisBrazil
  2. 2.School of Industrial and Systems EngineeringGeorgia Institute of TechnologyAtlantaUSA
  3. 3.IMPARio de JaneiroBrazil

Personalised recommendations