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A modified Solodov-Svaiter method for solving nonmonotone variational inequality problems

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Abstract

In a very interesting paper (SIAM J. Control Optim. 37(3): 765–776, 1999), Solodov and Svaiter introduced an effective projection algorithm with linesearch for finding a solution of a variational inequality problem (VIP) in Euclidean space. They showed that the iterative sequence generated by their algorithm converges to a solution of (VIP) under the main assumption that the cost mapping is pseudomonotone and continuous. In this paper, we propose to modify this algorithm for solving variational inequality problems in which the cost mapping is not required to be satisfied any pseudomonotonicity. Moreover, we do not use the embedded projection methods as in methods used in literature and the linesearch procedure is not necessary when the cost mapping is Lipschitz. Several numerical examples are also provided to illustrate the efficient of the proposed algorithms.

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References

  1. Ahn, B.H.: Iterative methods for linear complementarity problems with upper bounds on primary variables. Math. Program. 26, 295–315 (1983)

    Article  Google Scholar 

  2. Anh, P.K., Vinh, N.T.: Self-adaptive gradient projection algorithms for variational inequalities involving non-Lipschitz continuous operators. Numer. Algor. 81, 983–1001 (2019)

    Article  MathSciNet  Google Scholar 

  3. Bauschke, H.H., Combettes, P.L.: Convex analysis and monotone operator theory in hilbert spaces. Springer International Publishing (2017)

  4. Boyd, S., Vandenberghe, L.: Convex optimization. Cambridge University Press, Cambridge (2004)

    Book  Google Scholar 

  5. Burachik, R.S., Millan, R.D.: A projection algorithm for nonmonotone variational inequalities. Set Valued Var. Anal. 28, 149–166 (2020)

    Article  MathSciNet  Google Scholar 

  6. Burachik, R.S., Scheimberg, S.: A proximal point method for the variational inequality problem in Banach spaces. SIAM J. Control Optim. 39, 1633–1649 (2000)

    Article  MathSciNet  Google Scholar 

  7. Cai, G., Gibali, A., Iyiola, O.S., Shehu, Y.: A new double-projection method for solving variational inequalities in Banach spaces. J. Optim. Theory Appl. 178, 219–239 (2018)

    Article  MathSciNet  Google Scholar 

  8. Ceng, L.C., Hadjisavvas, N., Wong, N.C.: Strong convergence theorem by a hybrid extragradient-like approximation method for variational inequalities and fixed point problems. J. Glob. Optim. 46, 635–646 (2010)

    Article  MathSciNet  Google Scholar 

  9. Crouzeix, J.-P.: Pseudomonotone variational inequality problems: existence of solutions. Math. Program. 78, 305–314 (1997)

    MathSciNet  MATH  Google Scholar 

  10. Dinh, B.V., Kim, D.S.: Projection algorithms for solving nonmonotone equilibrium problems in Hilbert space. J. Comput. Appl. Math. 302, 106–117 (2016)

    Article  MathSciNet  Google Scholar 

  11. Dinh, B.V., Muu, L.D.: Algorithms for a class of bilevel programs involving pseudomonotone variational inequalities. Acta Math. Vietnam. 38, 529–540 (2013)

    Article  MathSciNet  Google Scholar 

  12. 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)

    MATH  Google Scholar 

  13. Harker, P.T., Pang, J.S.: Finite-dimensional variational inequality and nonlinear complementarity problems: a survey of theory, algorithms and applications. Math. Prog. 48, 161–220 (1990)

    Article  MathSciNet  Google Scholar 

  14. Hartman, P., Stampacchia, G.: On some nonlinear elliptic differential functional equations. Acta Mathematica. 115, 153–188 (1966)

    Article  Google Scholar 

  15. He, Y.R.: Solvability of the Minty variational inequality. J. Optim. Theory Appl. 174, 686–692 (2017)

    Article  MathSciNet  Google Scholar 

  16. Kanzow, C., Shehu, Y.: Strong convergence of a double projection-type method for monotone variational inequalities in Hilbert spaces. J. Fixed Point Theory Appl. 20. https://doi.org/10.1007/s11784-018-0531-8 (2018)

  17. Khanh, P.Q., Thong, D.V., Vinh, N.T.: Versions of the subgradient extragradient method for pseudomonotone variational inequalities, Acta Appl. Math. https://doi.org/10.1007/s10440-020-00335-9 (2020)

  18. Kinderlehrer, D., Stampacchia, G.: An introduction to variational inequalities and their applications. Academic Press (1987)

  19. Konnov, I.V.: Equilibrium models and variational inequalities. Elsevier, Amsterdam (2007)

    MATH  Google Scholar 

  20. Korpelevich, G.M.: The extragradient method for finding saddle points and other problems. Matekon. 12, 747–756 (1976)

    MathSciNet  MATH  Google Scholar 

  21. Iusem, A.N., Svaiter, B.F.: A variant of Korpelevich’s method for variational inequalities with a new search strategy. Optim. 42, 309–321 (1997)

    Article  MathSciNet  Google Scholar 

  22. Gafni, E.M., Bertsekas, D.P.: Two-metric projection methods for constrained optimization. SIAM J. Control Optim. 22, 936–964 (1984)

    Article  MathSciNet  Google Scholar 

  23. Malitsky, Y.V.: Projected reflected gradient methods for monotone variational inequalities. SIAM J Optim. 25, 502–520 (2015)

    Article  MathSciNet  Google Scholar 

  24. Solodov, M.V., Svaiter, B.F.: A new projection method for variational inequality problems. SIAM J. Control Optim. 37(3), 765–776 (1999)

    Article  MathSciNet  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  26. Sun, D.: A New Stepsize Skill for Solving a Class of Nonlinear Projection Equations. J. Comput. Math. 13, 357–368 (1995)

    MathSciNet  MATH  Google Scholar 

  27. Strodiot, J.J., Vuong, P.T., Van, N.T.T.: A class of shrinking projection extragradient methods for solving nonmonotone equilibrium problems in Hilbert spaces. J. Global Optim. 64, 159–178 (2016)

    Article  MathSciNet  Google Scholar 

  28. Thong, D.V., Hieu, D.V.: Inertial extragradient algorithms for strongly pseudomonotone variational inequalities. J. Comput. Appl. Math. 341, 80–98 (2018)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  31. Zarantonello, E.H.: Projections on convex sets in Hilbert space and spectral theory, contributions to nonlinear functional analysis, pp 237–424. Academic Press, New York (1971)

    Google Scholar 

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Acknowledgements

The authors are grateful to the editor and anonymous referees for their constructive suggestions which help them very much in revising their paper.

Funding

This research is supported by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 4/2020/STS02.

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Authors and Affiliations

  1. Department of Mathematics, Faculty of Information Technology, Le Quy Don Technical University, Hanoi, Vietnam

    Bui Van Dinh, Hy Duc Manh & Tran Thi Huyen Thanh

  2. Department of Optimization and Control Theory, Institue of Mathematics, VAST, Hanoi, Vietnam

    Bui Van Dinh

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  1. Bui Van Dinh
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  2. Hy Duc Manh
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  3. Tran Thi Huyen Thanh
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Correspondence to Bui Van Dinh.

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Van Dinh, B., Manh, H.D. & Thanh, T.T.H. A modified Solodov-Svaiter method for solving nonmonotone variational inequality problems. Numer Algor 90, 1715–1734 (2022). https://doi.org/10.1007/s11075-021-01248-w

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