Advertisement

Journal of Global Optimization

, Volume 65, Issue 3, pp 597–614 | Cite as

A relaxed-projection splitting algorithm for variational inequalities in Hilbert spaces

  • J. Y. Bello Cruz
  • R. Díaz Millán
Article

Abstract

We introduce a relaxed-projection splitting algorithm for solving variational inequalities in Hilbert spaces for the sum of nonsmooth maximal monotone operators, where the feasible set is defined by a nonlinear and nonsmooth continuous convex function inequality. In our scheme, the orthogonal projections onto the feasible set are replaced by projections onto separating hyperplanes. Furthermore, each iteration of the proposed method consists of simple subgradient-like steps, which does not demand the solution of a nontrivial subproblem, using only individual operators, which exploits the structure of the problem. Assuming monotonicity of the individual operators and the existence of solutions, we prove that the generated sequence converges weakly to a solution.

Keywords

Point-to-set operator Projection methods Relaxed method Splitting methods Variational inequality problem  Weak convergence 

Mathematics Subject Classification

90C47 49J35 

Notes

Acknowledgments

JYBC and RDM were partially supported by project CAPES-MES-CUBA 226/2012. JYBC was partially supported by CNPq grants 303492/2013-9, 474160/2013-0 and 202677/2013-3 and by project UNIVERSAL FAPEG/CNPq. RDM was supported by his scholarship for his doctoral studies, granted by CAPES. This work was completed while the first author was visiting the University of British Columbia. The author is very grateful for the warm hospitality. The authors would like to thank to Professor Dr. Heinz H. Bauschke, Professor Dr. Ole Peter Smith and anonymous referees whose suggestions helped us to improve the presentation of this paper.

References

  1. 1.
    Alber, YaI, Iusem, A.N., Solodov, M.V.: On the projected subgradient method for nonsmooth convex optimization in a Hilbert space. Math. Program. 81, 23–37 (1998)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Attouch, H., Czarnecki, M.-O., Peypouquet, J.: Coupling forward–backward with penalty schemes and parallel splitting for constrained variational inequalities. SIAM J. Optim. 21, 1251–1274 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bao, T.Q., Khanh, P.Q.: A projection-type algorithm for pseudomonotone nonlipschitzian multivalued variational inequalities. Nonconvex Opt. Appl. 77, 113–129 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bauschke, H.H., Combettes, P.L.: Convex Analysis and Monotone Operator Theory in Hilbert Spaces. Springer, New York (2011)CrossRefzbMATHGoogle Scholar
  5. 5.
    Bello Cruz, J.Y., Díaz Millán, R.: A direct splitting method for nonsmooth variational inequalities. J. Optim. Theory Appl. 161, 728–737 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Bello Cruz, J.Y., de Oliveira, W.: Level bundle-like algorithms for convex optimization. J. Global Optim. 59, 787–809 (2014)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Bello Cruz, J.Y., Iusem, A.N.: An explicit algorithm for monotone variational inequalities. Optimization 61, 855–871 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Bello Cruz, J.Y., Iusem, A.N.: A strongly convergent method for nonsmooth convex minimization in Hilbert spaces. Numer. Funct. Anal. Opt. 32, 1009–1018 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Bello Cruz, J.Y., Iusem, A.N.: Full convergence of an approximate projection method for nonsmooth variational inequalities. Math. Comput. Simul. 114, 2–13 (2015)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Bello Cruz, J.Y., Iusem, A.N.: Convergence of direct methods for paramonotone variational inequalities. Comput. Opt. Appl. 46, 247–263 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Bello Cruz, J.Y., Iusem, A.N.: A strongly convergent direct method for monotone variational inequalities in Hilbert spaces. Numer. Funct. Anal. Opt. 30, 23–36 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Boţ, R.I., Hendrich, C.: A Douglas-Rachford type primal-dual method for solving inclusions with mixtures of composite and parallel-sum type monotone operators. SIAM J. Optim. 23, 2541–2565 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Bruck, R.E.: On the weak convergence of an ergodic iteration for the solution of variational inequalities for monotone operators in Hilbert space. J. Math. Anal. Appl. 61, 159–164 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Burachik, R.S., Iusem, A.N.: Set-Valued Mappings and Enlargements of Monotone Operators. Springer, Berlin (2008)zbMATHGoogle Scholar
  15. 15.
    Burachik, R.S., Lopes, J.O., Svaiter, B.F.: An outer approximation method for the variational inequality problem. SIAM J. Control Opt. 43, 2071–2088 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Censor, Y., Gibali, A., Reich, S.: The subgradient extragradient method for solving variational inequalities in Hilbert space. J. Optim. Theory Appl. 148, 318–335 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Combettes, P.L.: Quasi-Fejérian analysis of some optimization algorithms. In: Inherently Parallel Algorithms in Feasibility and Optimization and Their Applications. Studies in Computational Mathematics 8, North-Holland, Amsterdam, pp. 115–152 (2001)Google Scholar
  18. 18.
    Combettes, P.L., Pesquet, J.-C.: Primal-dual splitting algorithm for solving inclusions with mixtures of composite, Lipschitzian, and parallel-sum type monotone operators. Set-Valued Var. Anal. 20, 307–330 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Eckstein, J., Svaiter, B.F.: General projective splitting methods for sums of maximal monotone operators. SIAM J. Control Opt. 48, 787–811 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Ermoliev, YuM: On the method of generalized stochastic gradients and quasi-Fejér sequences. Cybernetics 5, 208–220 (1969)CrossRefGoogle Scholar
  21. 21.
    Facchinei, F., Pang, J.S.: Finite-Dimensional Variational Inequalities and Complementarity Problems. Springer, Berlin (2003)zbMATHGoogle Scholar
  22. 22.
    Fang, S.C., Petersen, E.L.: Generalized variational inequalities. J. Optim. Theory Appl. 38, 363–383 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Fukushima, M.: A Relaxed projection for variational inequalities. Math. Program. 35, 58–70 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Gabay, D., Mercier, B.: A dual algorithm for the solution of nonlinear variational problems via finite element approximation. Comput. Math. Appl. 2, 17–40 (1976)CrossRefzbMATHGoogle Scholar
  25. 25.
    He, B.S.: A new method for a class of variational inequalities. Math. Program. 66, 137–144 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Iusem, A.N., Lucambio Pérez, L.R.: An extragradient-type method for non-smooth variational inequalities. Optimization 48, 309–332 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Iusem, A.N., Svaiter, B.F.: A variant of Korpelevich’s method for variational inequalities with a new search strategy. Optimization 42, 309–321 (1997) (Addendum Optimization 43, 85 (1998))Google Scholar
  28. 28.
    Iusem, A.N., Svaiter, B.F., Teboulle, M.: Entropy-like proximal methods in convex programming. Math. Oper. Res. 19, 790–814 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Kinderlehrer, D., Stampacchia, G.: An Introduction to Variational Inequalities and Their Applications. Academic Press, New York (1980)zbMATHGoogle Scholar
  30. 30.
    Konnov, I.V.: A combined relaxation method for variational inequalities with nonlinear constraints. Math. Program. 80, 239–252 (1998)MathSciNetzbMATHGoogle Scholar
  31. 31.
    Konnov, I.V.: Combined Relaxation Methods for Variational Inequalities. Springer, Berlin (2001)CrossRefzbMATHGoogle Scholar
  32. 32.
    Korpelevich, G.M.: The extragradient method for finding saddle points and other problems. Ekonomika I Matematcheskie Metody 12, 747–756 (1976)MathSciNetzbMATHGoogle Scholar
  33. 33.
    Lewis, A.S., Pang, J.-S.: Error Bounds for Convex Inequality Systems. Generalized Convexity, Generalized Monotonicity: Recent Results: Nonconvex Optimization and Its Applications, Vol. 27, pp. 75–110. Kluwer Academic Publications, Dordrecht (1998)Google Scholar
  34. 34.
    Lions, P.L., Mercier, B.: Splitting algorithms for the sum of two nonlinear operators. SIAM J. Numer. Anal. 16, 964–979 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Moudafi, A.: On the convergence of splitting proximal methods for equilibrium problems in Hilbert spaces. J. Math. Anal. Appl. 359, 508–513 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Nedic, A., Bertsekas, D.: Incremental subgradient methods for nondifferentiable optimization. SIAM J. Optim. 12, 109–138 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Nesterov, Yu.: Introductory Lectures on Convex Optimization: A Basic Course. Kluwer Academic Publishers, Norwel (2004)CrossRefzbMATHGoogle Scholar
  38. 38.
    Nesterov, Yu.: A method of solving a convex programming problem with convergence rate O(\(1/k^2\)). Soviet Math. Doklady 27, 372–376 (1983)MathSciNetzbMATHGoogle Scholar
  39. 39.
    Passty, G.B.: Ergodic convergence to a zero of the sum of monotone operators in Hilbert space. J. Math. Anal. Appl. 72, 383–390 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Polyak, B.T.: Minimization of unsmooth functionals. USSR Comput. Math. Math. Phys. 9, 14–29 (1969)CrossRefzbMATHGoogle Scholar
  41. 41.
    Rockafellar, R.T.: On the maximality of sums of nonlinear monotone operators. Trans. Am. Math. Soc. 149, 75–88 (1970)MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Rockafellar, R.T.: On the maximal monotonicity of subdifferential mapping. Pac. J. Math. 33, 209–216 (1970)MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Sien, D.: Computable error bounds for convex inequality systems in reflexive Banach spaces. SIAM J. Optim. 1, 274–279 (1997)MathSciNetzbMATHGoogle Scholar
  44. 44.
    Shih, M.H., Tan, K.K.: Browder–Hartmann–Stampacchia variational inequalities for multi-valued monotone operators. J. Math. Anal. Appl. 134, 431–440 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    Solodov, M.V., Svaiter, B.F.: A new projection method for monotone variational inequality problems. SIAM J. Control Opt. 37, 765–776 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  46. 46.
    Solodov, M.V., Tseng, P.: Modified projection-type methods for monotone variational inequalities. SIAM J. Control Opt. 34, 1814–1830 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  47. 47.
    Todd, M.J.: The Computations of Fixed Points and Applications. Springer, Berlin (1976)CrossRefGoogle Scholar
  48. 48.
    Tseng, P.: A modified forward–backward splitting method for maximal monotone mappings. SIAM J. Control Opt. 38, 431–446 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  49. 49.
    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
  50. 50.
    Zhang, H., Cheng, L.: Projective splitting methods for sums of maximal monotone operators with applications. J. Math. Anal. Appl. 406, 323–334 (2013)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Institute of Mathematics and StatisticsFederal University of GoiásGoiâniaBrazil
  2. 2.MathematicsFederal Institute of Education, Science and TechnologyGoiâniaBrazil

Personalised recommendations