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Convergence of direct methods for paramonotone variational inequalities

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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.

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  1. Alber, Y.I.: Recurrence relations and variational inequalities. Sov. Math. Dokl. 27, 511–517 (1983)

    Google Scholar 

  2. Alber, Ya.I., Iusem, A.N.: Extension of subgradient techniques for nonsmooth optimization in Banach spaces. Set-Valued Anal. 9, 315–335 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    MathSciNet  Google Scholar 

  4. Auslender, A., Teboulle, M.: Interior projection-like methods for monotone variational inequalities. Math. Program. 104, 39–68 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  7. Burachik, R.S., Iusem, A.N.: Set-Valued Mappings and Enlargements of Monotone Operators. Springer, Berlin (2008)

    Google Scholar 

  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)

    MathSciNet  Google Scholar 

  9. Facchinei, F., Pang, J.S.: Finite-dimensional Variational Inequalities and Complementarity Problems. Springer, Berlin (2003)

    Google Scholar 

  10. Fang, S.-C.: An iterative method for generalized complementarity problems. IEEE Trans. Autom. Control 25, 1225–1227 (1980)

    Article  MATH  Google Scholar 

  11. Fukushima, M.: An outer approximation algorithm for solving general convex programs. Oper. Res. 31, 101–113 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  12. Fukushima, M.: A relaxed projection for variational inequalities. Math. Program. 35, 58–70 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  13. Hartman, P., Stampacchia, G.: On some non-linear elliptic differential-functional equations. Acta Math. 115, 271–310 (1966)

    Article  MATH  MathSciNet  Google Scholar 

  14. Iusem, A.N.: On some properties of paramonotone operators. J. Convex Anal. 5, 269–278 (1998)

    MATH  MathSciNet  Google Scholar 

  15. Iusem, A.N., Lucambio Pérez, L.R.: An extragradient-type method for non-smooth variational inequalities. Optimization 48, 309–332 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  16. Korpelevich, G.M.: The extragradient method for finding saddle points and other problems. Ekon. Mat. Metody 12, 747–756 (1976)

    MATH  Google Scholar 

  17. Polyak, B.T.: Introduction to Optimization. Optimization Software, New York (1987)

    Google Scholar 

  18. Qu, B., Xiu, N.: A new halfspace-relaxation projection method for the split feasibility problem. Linear Algebra Appl. 428, 1218–1229 (2008)

    Article  MATH  MathSciNet  Google Scholar 

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Correspondence to J. Y. Bello Cruz.

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Bello Cruz, J.Y., Iusem, A.N. Convergence of direct methods for paramonotone variational inequalities. Comput Optim Appl 46, 247–263 (2010).

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