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Modified extragradient-like algorithms with new stepsizes for variational inequalities

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Abstract

The paper concerns with an algorithm for approximating solutions of a variational inequality problem involving a Lipschitz continuous and monotone operator in a Hilbert space. The algorithm uses a new stepsize rule which does not depend on the Lipschitz constant and without any linesearch procedure. The resulting algorithm only requires to compute a projection on feasible set and a value of operator over each iteration. The convergence and the convergence rate of the algorithm are established. Some experiments are performed to show the numerical behavior of the proposed algorithm and also to compare its performance with those of others.

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References

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

    Google Scholar 

  2. 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  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

  6. Fichera, G.: Sul problema elastostatico di Signorini con ambigue condizioni al contorno. Atti Accad. Naz. Lincei, VIII. Ser., Rend., Cl. Sci. Fis. Mat. Nat. 34, 138–142 (1963)

    MathSciNet  MATH  Google Scholar 

  7. Fichera, G.: Problemi elastostatici con vincoli unilaterali: il problema di Signorini con ambigue condizioni al contorno. Atti Accad. Naz. Lincei, Mem., Cl. Sci. Fis. Mat. Nat., Sez. I, VIII. Ser. 7, 91–140 (1964)

    MathSciNet  MATH  Google Scholar 

  8. Gibali, A.: A new Bregman projection method for solving variational inequalities in Hilbert spaces. Pure Appl. Funct. Anal. 3, 403–415 (2018)

    MathSciNet  Google Scholar 

  9. Gibali, A.: A new non-Lipschitzian projection method for solving variational inequalities in Euclidean spaces. J. Nonlinear Anal. Optim. 6, 41–51 (2015)

    MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  12. Hieu, D.V., Thong, D.V.: New extragradient-like algorithms for strongly pseudomonotone variational inequalities. J. Glob. Optim. 70, 385–399 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  13. Kanzow, C.: Some equation-based methods for the nonlinear complemantarity problem. Optim. Methods Softw. 3, 327–340 (1994)

    Article  Google Scholar 

  14. Kassay, G., Reich, S., Sabach, S.: Iterative methods for solving systems of variational inequalities in reflexive Banach spaces. SIAM J. Optim. 21, 1319–1344 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  15. Khanh, P.D.: A new extragradient method for strongly pseudomonotone variational inequalities. Numer. Funct. Anal. Optim. 37, 1131–1143 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  16. Khanh, P.D., Vuong, P.T.: Modified projection method for strongly pseudomonotone variational inequalities. J. Glob. Optim. 58, 341–350 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  17. Khobotov, E.N.: Modification of the extragradient method for solving variational inequalities and certain optimization problems. USSR Comput. Math. Math. Phys. 27, 120–127 (1989)

    Article  MATH  Google Scholar 

  18. Kinderlehrer, D., Stampacchia, G.: An Introduction to Variational Inequalities and Their Applications. Academic Press, New York (1980)

    MATH  Google Scholar 

  19. Konnov, I.V.: Combined Relaxation Methods for Variational Inequalities. Springer, Berlin (2000)

    MATH  Google Scholar 

  20. Konnov, I.V.: Equilibrium Models and Variational Inequalities. Elsevier, Amsterdam (2007)

    MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  22. Malitsky, Y.: Golden Ratio Algorithms for Variational Inequalities (2018). arXiv:1803.08832

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

    Article  MathSciNet  MATH  Google Scholar 

  24. Malitsky, Y.V., Semenov, V.V.: An extragradient algorithm for monotone variational inequalities. Cybern. Syst. Anal. 50, 271–277 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  25. Popov, L.D.: A modification of the Arrow–Hurwicz method for searching for saddle points. Mat. Zametki 28(5), 777–784 (1980)

    MathSciNet  MATH  Google Scholar 

  26. Semenov, V.V.: A version of the mirror descent method to solve variational inequalities. Cybern. Syst. Anal. 53, 234–243 (2017)

    Article  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  28. Sun, D.: A projection and contraction method for the nonlinear complementarity problems and its extensions. Math. Numer. Sin. 16, 183–194 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  29. Thong, D.V., Hieu, D.V.: Weak and strong convergence theorems for variational inequality problems. Numer. Algor. 78, 1045–1060 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  30. Tinti, F.: Numerical solution for pseudomonotone variational inequality problems by extragradient methods. Var. Anal. Appl. 79, 1101–1128 (2004)

    MathSciNet  MATH  Google Scholar 

  31. Vuong, P.T., Shehu, Y.: Convergence of an extragradient-type method for variational inequality with applications to optimal control problems. Numer. Algor. (2018). https://doi.org/10.1007/s11075-018-0547-6

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

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The authors would like to thank the Associate Editor and two anonymous referees for their valuable comments and suggestions which helped us very much in improving the original version of this paper. The first and third-named authors are supported by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under the project: 101.01-2017.315.

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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

    Dang Van Hieu

  2. Department of Mathematics, Vietnam National University, Hanoi, 334 Nguyen Trai, Thanh Xuan, Hanoi, Vietnam

    Pham Ky Anh

  3. TIMAS, Thang Long University, Hanoi, Vietnam

    Le Dung Muu

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  1. Dang Van Hieu
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  2. Pham Ky Anh
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  3. Le Dung Muu
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Correspondence to Dang Van Hieu.

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Van Hieu, D., Anh, P.K. & Muu, L.D. Modified extragradient-like algorithms with new stepsizes for variational inequalities. Comput Optim Appl 73, 913–932 (2019). https://doi.org/10.1007/s10589-019-00093-x

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