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Strong convergence of extragradient methods for solving bilevel pseudo-monotone variational inequality problems

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Abstract

In this paper, we propose two extragradient methods for finding an element of the set of solutions of the bilevel pseudo-monotone variational inequality problems in real Hilbert spaces. The advantage of proposed algorithms requires only one projection onto the feasible set. The strong convergence theorems are proved under mild conditions. Our results improve related results in the literature. Finally, some numerical experiments are presented to show the efficiency and advantages of the proposed algorithms.

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References

  1. Cegielski, A.: Iterative Methods for Fixed Point Problems in Hilbert Spaces Lecture Notes in Mathematics, vol. 2057. Springer, Berlin (2012)

    Google Scholar 

  2. Anh, P.N., Kim, J.K., Muu, L.D.: An extragradient algorithm for solving bilevel pseudomonotone variational inequalities. J. Glob. Optim. 52, 627–639 (2012)

    MathSciNet  MATH  Google Scholar 

  3. Antipin, A.S.: On a method for convex programs using a symmetrical modification of the Lagrange function. Ekon. Matematicheskie Metody 12, 1164–1173 (1976)

    Google Scholar 

  4. Baiocchi, C., Capelo, A.: Variational and Quasivariational Inequalities: Applications to Free Boundary Problems. Wiley, New York (1984)

    MATH  Google Scholar 

  5. Censor, Y., Gibali, A., Reich, S.: Extensions of Korpelevich’s extragradient method for the variational inequality problem in Euclidean space. Optimization 61, 1119–1132 (2011)

    MathSciNet  MATH  Google Scholar 

  6. Censor, Y., Gibali, A., Reich, S.: Strong convergence of subgradient extragradient methods for the variational inequality problem in Hilbert space. Optim. Meth. Softw. 26, 827–845 (2011)

    MathSciNet  MATH  Google Scholar 

  7. 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)

    MathSciNet  MATH  Google Scholar 

  8. Cottle, R.W., Yao, J.C.: Pseudo-monotone complementarity problems in Hilbert space. J. Optim. Theory Appl. 75, 281–295 (1992)

    MathSciNet  MATH  Google Scholar 

  9. Daniele, P., Giannessi, F., Maugeri, A.: Equilibrium Problems and Variational Models. Kluwer Academic, Dordrecht (2003)

    MATH  Google Scholar 

  10. Denisov, S.V., Semenov, V.V., Chabak, L.M.: Convergence of the modified extragradient method for variational inequalities with non-Lipschitz operators. Cybern. Syst. Anal. 51, 757–765 (2015)

    MathSciNet  MATH  Google Scholar 

  11. Ding, X.P.: Auxiliary principle and algorithm for mixed equilibrium problems and bilevel mixed equilibrium problems in Banach spaces. J. Optim. Theory Appl. 146, 347–357 (2010)

    MathSciNet  MATH  Google Scholar 

  12. Ding, X.P.: Existence and algorithm of solutions for mixed equilibrium problems and bilevel mixed equilibrium problems in Banach spaces. Acta Math. Sin. Engl. Ser. 28(3), 503–514 (2011)

    MathSciNet  MATH  Google Scholar 

  13. Dong, Q.L., Cho, Y.J., Rassias, Th.M.: The projection and contraction methods for finding common solutions to variational inequality problems. Optim. Lett. 12(8), 1871–1896 (2018)

    MathSciNet  MATH  Google Scholar 

  14. Dong, Q.L., Lu, Y.Y., Yang, J.: The extragradient algorithm with inertial effects for solving the variational inequality. Optim 65, 2217–2226 (2016)

    MathSciNet  MATH  Google Scholar 

  15. Facchinei, F., Pang, J.S.: Finite-Dimensional Variational Inequalities and Complementarity Problems. Springer Series in Operations Research, Vols I And II. Springer, New York (2003)

    Google Scholar 

  16. Giannessi, F., Maugeri, A., Pardalos, P.M.: Equilibrium Problems: Nonsmooth Optimization and Variational Inequality Models. Kluwer Academic, Dordrecht (2004)

    MATH  Google Scholar 

  17. Glackin, J., Ecker, J.G., Kupferschmid, M.: Solving bilevel linear programs using multiple objective linear programming. J. Optim. Theory Appl. 140, 197–212 (2009)

    MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

  19. Harker, P.T., Pang, J.S.: A damped-Newton method for the linear complementarity problem. Lect. Appl. Math. 26, 265–284 (1990)

    MathSciNet  MATH  Google Scholar 

  20. He, B.S., Liao, L.Z.: Improvements of some projection methods for monotone nonlinear variational inequalities. J. Optim. Theory Appl. 112, 111–128 (2002)

    MathSciNet  MATH  Google Scholar 

  21. Hu, X., Wang, J.: Solving pseudo-monotone variational inequalities and pseudo-convex optimization problems using the projection neural network. IEEE Trans. Neural Netw. 17, 1487–1499 (2006)

    Google Scholar 

  22. Karamardian, S., Schaible, S.: Seven kinds of monotone maps. J. Optim. Theory Appl. 66, 37–46 (1990)

    MathSciNet  MATH  Google Scholar 

  23. Konnov, I.V.: Combined Relaxation Methods for Variational Inequalities. Springer, Berlin (2001)

    MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  25. Kraikaew, R., Saejung, S.: On a hybrid extragradient-viscosity method for monotone operators and fixed point problems. Numer. Funct. Anal. Optim. 35, 32–49 (2014)

    MathSciNet  MATH  Google Scholar 

  26. Kraikaew, R., Saejung, S.: Strong convergence of the Halpern subgradient extragradient method for solving variational inequalities in Hilbert spaces. J. Optim. Theory Appl. 163, 399–412 (2014)

    MathSciNet  MATH  Google Scholar 

  27. Luo, Z.Q., Pang, J.S., Ralph, D.: Mathematical Programs with Equilibrium Constraints. Cambridge University Press, Cambridge (1996)

    MATH  Google Scholar 

  28. Moudafi, A.: Proximal methods for a class of bilevel monotone equilibrium problems. J. Glob. Optim. 47, 287–292 (2010)

    MathSciNet  MATH  Google Scholar 

  29. Muu, L.D., Quy, N.V.: On existence and solution methods for strongly pseudomonotone equilibrium problems. Vietnam J. Math. 43, 229–238 (2015)

    MathSciNet  MATH  Google Scholar 

  30. Saejung, S., Yotkaew, P.: Approximation of zeros of inverse strongly monotone operators in Banach spaces. Nonlinear Anal. 75, 742–750 (2012)

    MathSciNet  MATH  Google Scholar 

  31. Shehu, Y., Dong, Q.L., Jiang, D.: Single projection method for pseudo-monotone variational inequality in Hilbert spaces. Optimization 68, 385–409 (2019)

    MathSciNet  MATH  Google Scholar 

  32. Solodov, M.V.: An explicit descent method for bilevel convex optimization. J. Convex Anal. 14, 227–237 (2007)

    MathSciNet  MATH  Google Scholar 

  33. Solodov, M.V., Svaiter, B.F.: A new projection method for variational inequality problems. SIAM J. Control Optim. 37, 765–776 (1999)

    MathSciNet  MATH  Google Scholar 

  34. Thong, D.V., Hieu, D.V.: Modified subgradient extragradient method for variational inequality problems. Numer. Algorithm. 79, 597–610 (2017)

    MathSciNet  MATH  Google Scholar 

  35. Thong, D.V., Hieu, D.V.: Weak and strong convergence theorems for variational inequality problems. Numer. Algorithm. 78, 1045–1060 (2017)

    MathSciNet  MATH  Google Scholar 

  36. Trujillo, C.R., Zlobec, S.: Bilevel convex programming models. Optimization. 58, 1009–1028 (2009)

    MathSciNet  MATH  Google Scholar 

  37. Tseng, P.: A modified forward-backward splitting method for maximal monotone mappings. SIAM J. Control Optim. 38, 431–446 (2000)

    MathSciNet  MATH  Google Scholar 

  38. Vuong, P.T.: On the weak convergence of the extragradient method for solving pseudomonotone variational inequalities. J. Optim. Theory Appl. 176, 399–409 (2018)

    MathSciNet  MATH  Google Scholar 

  39. Xu, M.H., Li, M., Yang, C.C.: Neural networks for a class of bi-level variational inequalities. J. Glob. Optim. 44, 535–552 (2009)

    MathSciNet  MATH  Google Scholar 

  40. Yang, J., Liu, H.: Strong convergence result for solving monotone variational inequalities in Hilbert space. Numer. Algorithms 80(3), 741–752 (2019)

    MathSciNet  MATH  Google Scholar 

  41. Yang, J., Liu, H., Liu, Z.: Modified subgradient extragradient algorithms for solving monotone variational inequalities. Optimization 67(12), 2247–2258 (2018)

    MathSciNet  MATH  Google Scholar 

  42. Yao, Y., Marino, G., Muglia, L.: A modified Korpelevich’s method convergent to the minimum-norm solution of a variational inequality. Optimization 63, 559–569 (2014)

    MathSciNet  MATH  Google Scholar 

  43. Zegeye, H., Shahzad, N., Yao, Y.: Minimum-norm solution of variational inequality and fixed point problem in Banach spaces. Optimization 64, 453–471 (2015)

    MathSciNet  MATH  Google Scholar 

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Acknowledgments

The authors would like to thank Dr. Aviv Gibali and two anonymous reviewers for their comments on the manuscript which helped us very much in improving and presenting the original version of this paper.

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Authors and Affiliations

  1. Applied Analysis Research Group, Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam

    Duong Viet Thong

  2. Institute of Fundamental and Applied Sciences, Duy Tan University, 03 Quang Trung, Da Nang, Vietnam

    Nguyen Anh Triet

  3. Tianjin Key Laboratory for Advanced Signal Processing and College of Science, Civil Aviation University of China, Tianjin, 300300, China

    Xiao-Huan Li & Qiao-Li Dong

Authors
  1. Duong Viet Thong
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  2. Nguyen Anh Triet
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  3. Xiao-Huan Li
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  4. Qiao-Li Dong
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Corresponding author

Correspondence to Nguyen Anh Triet.

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Dedicated to Professor Pham Ky Anh on the Occasion of his 70th Birthday.

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Thong, D.V., Triet, N.A., Li, XH. et al. Strong convergence of extragradient methods for solving bilevel pseudo-monotone variational inequality problems. Numer Algor 83, 1123–1143 (2020). https://doi.org/10.1007/s11075-019-00718-6

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  • DOI: https://doi.org/10.1007/s11075-019-00718-6

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