Computational Optimization and Applications

, Volume 46, Issue 2, pp 247–263 | Cite as

Convergence of direct methods for paramonotone variational inequalities

  • J. Y. Bello Cruz
  • A. N. Iusem


We analyze one-step direct methods for variational inequality problems, establishing convergence under paramonotonicity of the operator. Previous results on the method required much more demanding assumptions, like strong or uniform monotonicity, implying uniqueness of solution, which is not the case for our approach.


Monotone variational inequalities Maximal monotone operators Projection method 


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  1. 1.
    Alber, Y.I.: Recurrence relations and variational inequalities. Sov. Math. Dokl. 27, 511–517 (1983) Google Scholar
  2. 2.
    Alber, Ya.I., Iusem, A.N.: Extension of subgradient techniques for nonsmooth optimization in Banach spaces. Set-Valued Anal. 9, 315–335 (2001) zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Alber, Ya.I., 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) MathSciNetGoogle Scholar
  4. 4.
    Auslender, A., Teboulle, M.: Interior projection-like methods for monotone variational inequalities. Math. Program. 104, 39–68 (2005) zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Bruck, R.E.: An iterative solution of a variational inequality for certain monotone operators in a Hilbert space. Bull. Am. Math. Soc. 81, 890–892 (1975) zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Burachik, R., Graña Drummond, L.M., Iusem, A.N., Svaiter, B.F.: Full convergence of the steepest descent method with inexact line searches. Optimization 32, 137–146 (1995) zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Burachik, R.S., Iusem, A.N.: Set-Valued Mappings and Enlargements of Monotone Operators. Springer, Berlin (2008) Google Scholar
  8. 8.
    Censor, Y., Iusem, A.N., Zenios, S.A.: An interior point method with Bregman functions for the variational inequality problem with paramonotone operators. Math. Program. 81, 373–400 (1998) MathSciNetGoogle Scholar
  9. 9.
    Facchinei, F., Pang, J.S.: Finite-dimensional Variational Inequalities and Complementarity Problems. Springer, Berlin (2003) Google Scholar
  10. 10.
    Fang, S.-C.: An iterative method for generalized complementarity problems. IEEE Trans. Autom. Control 25, 1225–1227 (1980) zbMATHCrossRefGoogle Scholar
  11. 11.
    Fukushima, M.: An outer approximation algorithm for solving general convex programs. Oper. Res. 31, 101–113 (1983) zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Fukushima, M.: A relaxed projection for variational inequalities. Math. Program. 35, 58–70 (1986) zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Hartman, P., Stampacchia, G.: On some non-linear elliptic differential-functional equations. Acta Math. 115, 271–310 (1966) zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Iusem, A.N.: On some properties of paramonotone operators. J. Convex Anal. 5, 269–278 (1998) zbMATHMathSciNetGoogle Scholar
  15. 15.
    Iusem, A.N., Lucambio Pérez, L.R.: An extragradient-type method for non-smooth variational inequalities. Optimization 48, 309–332 (2000) zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Korpelevich, G.M.: The extragradient method for finding saddle points and other problems. Ekon. Mat. Metody 12, 747–756 (1976) zbMATHGoogle Scholar
  17. 17.
    Polyak, B.T.: Introduction to Optimization. Optimization Software, New York (1987) Google Scholar
  18. 18.
    Qu, B., Xiu, N.: A new halfspace-relaxation projection method for the split feasibility problem. Linear Algebra Appl. 428, 1218–1229 (2008) zbMATHCrossRefMathSciNetGoogle Scholar

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© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  1. 1.Instituto de Matemática Pura e AplicadaRio de JaneiroBrazil

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