Numerical Algorithms

, Volume 78, Issue 4, pp 1111–1127 | Cite as

Two algorithms for solving systems of inclusion problems

  • R. Díaz Millán
Original Paper


The goal of this paper is to present two algorithms for solving systems of inclusion problems, with all components of the systems being a sum of two maximal monotone operators. The algorithms are variants of the forward-backward splitting method and one being a hybrid with the alternating projection method. They consist of approximating the solution sets involved in the problem by separating half-spaces which is a well-studied strategy. The schemes contain two parts, the first one is an explicit Armijo-type search in the spirit of the extragradient-like methods for variational inequalities. The second part is the projection step, this being the main difference between the algorithms. While the first algorithm computes the projection onto the intersection of the separating half-spaces, the second chooses one component of the system and projects onto the separating half-space of this case. In the iterative process, the forward-backward operator is computed once per inclusion problem, representing a relevant computational saving if compared with similar algorithms in the literature. The convergence analysis of the proposed methods is given assuming monotonicity of all operators, without Lipschitz continuity assumption. We also present some numerical experiments.


Armijo-type search Maximal monotone operators Forward-backward Alternating projection Systems of inclusion problems 

Mathematics Subject Classification (2010)

93B40 65K15 68W25 47H05 49J40 


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The author was partially supported by CNPq grant 200427/2015-6. This work was concluded while the author was visiting the School of Information Technology and Mathematical Sciences at the University of South Australia. The author would like to thank the great hospitality received during his visit, particularly to Regina S. Burachik and C. Yalçin Kaya. The author would like to express his gratitude to two anonymous referees for their valuable comments and suggestions that are very helpful to improve this paper.


  1. 1.
    Al-Homidan, S., Alshahrani, M., Ansari, Q.H.: System of nonsmooth variational inequalities with applications. Optimization 64(5), 1211–1218 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bauschke, H.H., Borwein, J.M.: On projection algorithms for solving convex feasibility problems. SIAM Rev. 38, 367–426 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bauschke, H.H., Combettes, P.L.: Convex analysis and monotone operator theory in hilbert spaces. Springer (2011)Google Scholar
  4. 4.
    Bello Cruz, J.Y., Díaz Millán, R.: A variant of forward-backward splitting method for the sum of two monotone operators with a new search strategy. Optimization 64(7), 1471–1486 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Browder, F.E.: Convergence theorems for sequences of nonlinear operators in Banach spaces. Math. Z. 100, 201–225 (1967)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Censor, Y., Gibali, A., Reich, S.: A von Neumann alternating method for finding common solutions to variational inequalities. Nonlinear Analysis Series A: Theory, Methods and Applications 75, 4596–4603 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Censor, Y., Gibali, A., Reich, S., Sabach, S.: Common solutions to variational inequalities. Set-Valued and Variational Analysis 20, 229–247 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Censor, Y., Gibali, A., Reich, S.: Algorithms for the split variational inequality problem. Numerical Algorithms 59, 301–323 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Combettes, P.L.: Fejér monotonicity in convex optimization. Encyclopedia of Optimization 1016–1024 (2009)Google Scholar
  10. 10.
    Díaz Millán, R.: On several algorithms for variational inequality and inclusion problems. PhD thesis, Federal University of Goiás, Goiânia, GO. Institute of Mathematic and Statistic, IME-UFG (2015)Google Scholar
  11. 11.
    Douglas, J., Rachford Jr., H.H.: On the numerical solution of heat conduction problems in two or three space variables. Trans. Amer. Math. Soc. 82, 421–439 (1956)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Eckstein, J.: Splitting methods for monotone operators, with applications to parallel optimization. PhD Thesis, Massachusetts Institute of Techonology, Cambridge, MA. Report LIDS-TH-1877, Laboratory for Information and Decision Systems, M.I.T (1989)Google Scholar
  13. 13.
    Eslamian, M., Saejung, S., Vahidi, J.: Common solutions of a system of variational inequality problems. UPB Scientific Bulletin Series A: Applied Mathematics and Physics Seria A 77(1), 55–62 (2015)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Harker, P.T., Pang, J.S.: A damped-newton method for the linear complementarity problem. Lect. Appl. Math. 26, 265–284 (1990)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Iusem, A.N., Svaiter, B.F., Teboulle, M.: Entropy-like proximal methods in convex programming. Math. Oper. Res. 19, 790–814 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Kopecká, E., Reich, S.: Another note on the von Neumann alternating projections algorithm. Journal of Nonlinear and Convex Analysis 11(3), 455–460 (2010)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Kopecká, E., Reich, S.: A note on the von Neumann alternating projections algorithm. Journal of Nonlinear Convex Analysis 5, 379–386 (2004)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Konnov, I.V.: On systems of variational inequalities. Russian Mathematics 41(12), 79–88 (1997)MathSciNetGoogle Scholar
  19. 19.
    Konnov, I.V.: Splitting-type method for systems of variational inequalities. Comput. Oper. Res. 33, 520–534 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Minty, G.: On the maximal domain of a “monotone” function. Mich. Math. J. 8, 135–137 (1961)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Minty, G.: Monotone (nonlinear) operators in Hilbert Space. Duke Mathematical Journal 29, 341–346 (1962)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Rosasco, L., Villa, S., Vu, B.C.: Stochastic forward-backward splitting for monotone inclusions. J. Optim. Theory Appl. 169(2), 388–406 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Semenov, V.V.: Hybrid splitting methods for the system of operator inclusions with monotone operators. Cybern. Syst. Anal. 50, 741–749 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Villa, S., Salzo, S., Baldassarre, L., Verri, A.: Accelerated and inexact forward-backward algorithms. SIAM J. Optim. 23(3), 1607–1633 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Tseng, P.: A modified forward-backward splitting method for maximal monotone mappings. SIAM J. Control. Optim. 38, 431–446 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Van Hieu, D., Anh, P.K., Muu, L.D.: Modified hybrid projection methods for finding common solutions to variational inequality problems. Comput. Optim. Appl. (2016)Google Scholar
  27. 27.
    Zarantonello, E.H.: Projections on convex sets in Hilbert space and spectral theory. In: Zarantonello, E (ed.) Contributions to Nonlinear Functional Analysis, pp 237–424. Academic Press, New York (1971)Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.Federal Institute of GoiásGoiâniaBrazil

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