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Approximately solving multi-valued variational inequalities by using a projection and contraction algorithm

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Abstract

A projection and contraction algorithm for solving multi-valued variational inequalities is proposed. The algorithm is proved to converge globally to a solution of a given multi-valued variational inequality under standard conditions. We present an analysis of the convergence rate. Finally, preliminary computational experiments illustrate the advantage of the proposed algorithm.

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References

  1. Aubin, J.P., Ekeland, I.: Applied Nonlinear Analysis. Wiley, New York (1984)

  2. Bao, T.Q., Khanh, P.Q.: A projection-type algorithm for pseudomonotone nonlipschitzian multivalued variational inequalities. In: Eberhard, A., Vvas, N.H., Luc, D.T. (eds.) Generalized Convexity, Generalized Monotonicity and Applications. Springer, New York (2005)

  3. Bauschke, H.H., Combettes, P.L.: A weak-to-strong convergence principle for Fejér-monotone methods in Hilbert spaces. Math. Oper. Res. 26, 248–264 (1999)

  4. Bauschke, H.H., Combettes, P.L.: Convex Analysis and Monotone Operator Theory in Hilbert Spaces. Springer, Berlin (2011)

  5. Bauschke, H.H., Matoušková, E., Reich, S.: Projection and proximal point methods: convergence results and counterexamples. Nonlinear Anal. 56, 715–738 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  6. Bnouhachem, A.: A self-adaptive method for solving general mixed variational inequalities. J. Math. Anal. Appl. 309, 136–150 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  7. Browder, F.E.: Multi-valued monotone nonlinear mappings duality mappings in Banach spaces. Trans. Amer. Math Soc. 118, 338–351 (1965)

    Article  MATH  MathSciNet  Google Scholar 

  8. Bruck, R.E., Reich, S.: Nonexpansive projections resolvents of accretive operators in Banach spaces Houston. J. Math. 3, 459–470 (1977)

    MATH  MathSciNet  Google Scholar 

  9. Cai, X., Gu, G., He, B.: On the O(1/t) convergence rate of the projection and contraction methods for variational inequalities with Lipschitz continuous monotone operators. Comput. Optim. Appl. 57, 339–363 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  10. Censor, Y., Gibali, A., Reich, S.: Extensions of Korpelevich’s extragradient method for solving the variational inequality problem in Euclidean space. Optimization 61(9), 1119–1132 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  11. 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)

    Article  MATH  MathSciNet  Google Scholar 

  12. Dong, Q.L., Yang, J., Yuan, H.B.: The projection and contraction algorithm for solving variational inequality problems in Hilbert spaces. J. Nonlinear Convex A. Accepted

  13. Facchinei, F., Pang, J.S.: Finite-dimensional variational inequalities and complementarity problems Volumes I and II. Springer, New York (2003)

    MATH  Google Scholar 

  14. Fang, C., Chen, S.: A subgradient extragradient algorithm for solving multi-valued variational inequality. Appl. Math. Comput. 229, 123–130 (2014)

    MATH  MathSciNet  Google Scholar 

  15. Fichera, G.: Problemi Elastostatici con Vincoli Unilaterali: il Problema di Signorini con Ambingue Condizioni al Contorno. Mem. Accad. Nax. Lincei, Ser. 8(7), 91–140 (1964)

    MATH  Google Scholar 

  16. Goebel, K., Reich, S.: Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings. Marcel Dekker, New York (1984)

  17. He, B.S.: A class of projection and contraction methods for monotone variational inequalities. Appl. Math. Opt. 35, 69–76 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  18. Hiriart-Urruty, J.B., Lemaréchal, C.: Fundamentals of Convex Analysis. Springer, Berlin (2001)

  19. Huebner, E., Tichatschke, R.: Relaxed proximal point algorithms for variational inequalities with multi-valued operators. Optim. Method Softw. 23, 847–877 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  20. Karamardian, S.: Complementarity problems over cones with monotone and pseudomonotone maps. J. Optim. Theory Appl. 18(4), 445–454 (1976)

    Article  MATH  MathSciNet  Google Scholar 

  21. Kiwiel, K.C.: Proximity control in bundle methods for convex nondifferentiable minimization. Math. Program. Ser. A 46(1), 105–122 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  22. Konnov, I.V.: A combined relaxation method for variational inequalities with nonlinear constraints. Math. Program. 80, 239–252 (1998)

    MATH  MathSciNet  Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

  24. Kulikov, A.N., Fazylov, V.R.: A finite solution method for systems of convex inequalities. Sov. Math. (Izvestiya VUZMatematika) 28(11), 75–80 (1984)

    MATH  Google Scholar 

  25. Lemarechal, C., Sagastizabal, C.: Variable metric bundle methods: from conceptual to implementable forms. Math. Program. Ser. B 76(3), 393–410 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  26. Martinet, B.: Regularisation d’inequations variationnelles par approximations successives. Rev. Franc. Autom. Inform. Rech. Oper. 4, 154–158 (1970)

    MATH  Google Scholar 

  27. Noor, M.A.: Some developments in general variational inequalities. Appl. Math. Comput. 152, 199–277 (2004)

    MATH  MathSciNet  Google Scholar 

  28. Polyak, B.T.: Introduction to optimization. Optimization Software Inc., Publications Division, New York (1987)

    MATH  Google Scholar 

  29. Rockafellar, R.T.: Monotone operators and the proximal point algorithm. SIAM J. Control Optim. 14(5), 877–898 (1976)

    Article  MATH  MathSciNet  Google Scholar 

  30. Schramm, H., Zowe, J.: A version of the bundle idea for minimizing a nonsmooth function: conceptual idea, convergence analysis, numerical results. SIAM J. Optim. 2(1), 121–152 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  31. Stampacchia, G.: Formes bilinéaires coercitives sur les ensembles convexes. C. R. Acad. Sci. Paris 258, 4413–4416 (1964)

    MATH  MathSciNet  Google Scholar 

  32. Sun, D.F.: A class of iterative methods for solving nonlinear projection equations. J. Optim. Theory Appl. 91, 123–140 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  33. Xia, F.Q., Huang, N.J.: A projection-proximal point algorithm for solving generalized variational inequalities. J. Optim. Theory Appl. 150, 98–117 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  34. Xiu, N., Zhang, J.: Some recent advances in projection-type methods for variational inequalities. J. Comput. Appl. Math. 152, 559–585 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  35. Yao, J.C.: Strong convergence theorems for strictly pseudocontractive mappings of browder-CPetryshyn type, Taiwan. J. Math. 3, 837–850 (2006)

    MATH  Google Scholar 

  36. Yao, J.C.: A proximal method for pseudomonotone type variational-like inequalities, Taiwan. J. Math. 2, 497–514 (2006)

    MATH  Google Scholar 

  37. Ye, M., He, Y.: A double projection method for solving variational inequalities without monotonicity. Comput. Optim. Appl. 60, 141–150 (2015)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. College of Science, Civil Aviation University of China, Tianjin, 300300, China

    Qiao-Li Dong, Yan-Yan Lu & Songnian He

  2. Tianjin Key Lab for Advanced Signal Processing, Civil Aviation University of China, Tianjin, 300300, China

    Qiao-Li Dong, Jinfeng Yang & Songnian He

Authors
  1. Qiao-Li Dong
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  2. Yan-Yan Lu
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  3. Jinfeng Yang
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  4. Songnian He
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Corresponding author

Correspondence to Jinfeng Yang.

Additional information

Supported by National Natural Science Foundation of China (No. 61379102) and Open Fund of Tianjin Key Lab for Advanced Signal Processing (No. 2016ASP-TJ01)

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Dong, QL., Lu, YY., Yang, J. et al. Approximately solving multi-valued variational inequalities by using a projection and contraction algorithm. Numer Algor 76, 799–812 (2017). https://doi.org/10.1007/s11075-017-0283-3

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  • DOI: https://doi.org/10.1007/s11075-017-0283-3

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