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A New Method for Solving Variational Inequalities and Fixed Points Problems of Demi-Contractive Mappings in Hilbert Spaces

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Abstract

In this paper, an efficient algorithm for solving variational inequalities and fixed points problems of demi-contractive mappings is proposed in Hilbert spaces. The algorithm uses variable stepsizes which are updated at each iteration by a cheap computation without any linesearch procedure. Under the assumption that the mapping is pseudomonotone and without prior knowledge of the Lipschitz constant of the underlying operator, the sequence generated by the algorithm is strongly convergent to a common element of the set of fixed points of a demi-contractive mapping and the solution set of variational inequalities. 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. Facchinei, F., Pang, J.S.: Finite-dimensional Variational Inequalities and Complementarity Problems, vol. I. Springer, New York (2003)

    MATH  Google Scholar 

  2. Iusem, A.N., Svaiter, B.F.: A variant of Korpelevich’s method for variational inequalities with a new search strategy. Optimization 42(4), 309–321 (1997)

    Article  MathSciNet  Google Scholar 

  3. 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 

  4. Ye, M.L., He, Y.R.: A double projection method for solving variational inequalities without mononicity. Comput. Optim. Appl. 60(1), 141–150 (2015)

    Article  MathSciNet  Google Scholar 

  5. Fang, C.J., He, Y.R.: An extragradient method for generalized variational inequality. Pac. J. Optim. 9, 47–59 (2013)

    MathSciNet  MATH  Google Scholar 

  6. 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  Google Scholar 

  7. Sibony, M.: Methodes iteratives pour les equation set enequalitions aux derivees partielles nonlinearesde type monotone. Calcolo 7, 65–183 (1970)

    Article  MathSciNet  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  9. Antipin, A.S.: On a method for convex programs using a symmetrical modification of the Lagrange function. Ekonomika i Matematicheskie Metody 12(6), 1164–1173 (1976)

    Google Scholar 

  10. Censor, Y., Gibali, A., Reich, S.: The subgradient extragradient method for solving variational inequalities in Hilbert spaces. J. Optim. Theory Appl. 148, 318–335 (2011)

    Article  MathSciNet  Google Scholar 

  11. Thong, D.V., Vinh, N.T., Cho, Y.J.: Accelerated subgradient extragradient methods for variational inequality problems. J. Sci. Comput. 80, 1436–1462 (2019)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  13. Hieu, D.V., Anh, P.K., Muu, L.D.: Modified extragradient-like algorithms with new stepsizes for variational inequalities. Comput. Optim. Appl. 73, 913–932 (2019)

    Article  MathSciNet  Google Scholar 

  14. Hieu, D.V., Cho, Y.J., Xiao, Y.B.: Golden ratio algorithms with new stepsize rules for variational inequalities. Math. Methods Appl. 42, 6067–6082 (2019)

    Article  MathSciNet  Google Scholar 

  15. Hieu, D.V., Cho, Y.J., Xiao, Y.B.: Modified extragradient algorithms for solving equilibrium problems. Optimization 67, 2003–2029 (2018)

    Article  MathSciNet  Google Scholar 

  16. Wang, Z.B., Chen, Z.Y., Xiao, X.B., Zhang, C.: A new projection-type method for solving multi-valued mixed variational inequalities without monotonicity. Appl. Anal. 99(9), 1453–1466 (2020)

    Article  MathSciNet  Google Scholar 

  17. Takahashi, W.: Convex Analysis and Approximation of Fixed Points. Yokohama Publishers, Yokohama (2000)

    MATH  Google Scholar 

  18. Yamada, I.: The hybrid steepest descent method for the variational inequality over the intersection of fixed point sets of nonexpansive mappings. in Inherently Parallel Algorithms in Feasibility and Optimization and Their Applications, D. Butnariu, Y. Censor, and S. Reich, eds., Elsevier, Amsterdam, pp. 473–504 (2001)

  19. Zeidler, E.: Nonlinear Functional Analysis and Its Applications. Springer, New York (1985)

    Book  Google Scholar 

  20. Mainge, P.E.: A hybrid extragradient-viscosity method for monotone operators and fixed point problems. SIAM J. Control Optim. 47(3), 1499–1515 (2008)

    Article  MathSciNet  Google Scholar 

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

    Article  Google Scholar 

  22. Thong, D.V., Hieu, D.V.: Inertial subgradient extragradient algorithms with line-search process for solving variational inequality problems and fixed point problems. Numer. Algor. 80(4), 1283–1307 (2019)

    Article  MathSciNet  Google Scholar 

  23. Gibali, A., Shehu, Y.: An efficient iterative method for finding common fixed point and variational inequalities in Hilbert spaces. Optimization 68(1), 13–32 (2019)

    Article  MathSciNet  Google Scholar 

  24. Iiduka, H., Yamada, I.: A use of conjugate gradient direction for the convex optimization problem over the fixed point set of a nonexpansive mapping. SIAM J. Optim. 19, 1881–1893 (2008)

    Article  MathSciNet  Google Scholar 

  25. Zhang, L., Fang, C., Chen, S.: An inertial subgradient-type method for solving single-valued variational inequalities and fixed point problems. Numer. Algor. 79, 941–956 (2018)

    Article  MathSciNet  Google Scholar 

  26. Vuong, P.T., Shehu, Y.: Convergence of an extragradient-type method for variational inequality with applications to optimal control problems. Numer. Algor. 81(1), 269–291 (2018)

    Article  MathSciNet  Google Scholar 

  27. Malitsky, Y.: Golden ratio algorithms for variational inequalities. Math. Prog. https://doi.org/10.1007/s10107-019-01416-w

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

    MATH  Google Scholar 

  29. Zarantonello, E.H.: Projections on Convex Sets in Hilbert space and Spectral theory. In: Zarantonello, E.H. (ed.) Contributions to Nonlinear Functional Analysis. Academic Press, New York (1971)

    MATH  Google Scholar 

  30. Xu, H.K.: Iterative algorithms for nonlinear operators. J. Lond. Math. Soc. 66(1), 240–256 (2002)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  32. Mashreghi, J., Nasri, M.: Forcing strong convergence of Korpelevich\(!^{-}\) method in Banach spaces with its applications in game theory. Nonlinear Anal. 72, 2086–2099 (2010)

    Article  MathSciNet  Google Scholar 

  33. Dong, Q.L., Lu, Y.Y., Yang, J.: The extragradient algorithm with inertial effects for solving the variational inequality. Optimization 65(12), 2217–2226 (2016)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  35. Malitsky, Y.V., Semenov, V.V.: A hybrid method without extrapolation step for solving variational inequality problems. J. Global Optim. 61, 193–202 (2015)

    Article  MathSciNet  Google Scholar 

  36. 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)

    Article  Google Scholar 

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Authors and Affiliations

  1. School of Mathematics, Southwest Jiaotong University, Chengdu, 611756, Sichuan, People’s Republic of China

    Xue Chen, Zhong-bao Wang & Zhang-you Chen

  2. National Engineering Laboratory of Integrated Transportation Big Data Application Technology, Chengdu, 611756, People’s Republic of China

    Xue Chen, Zhong-bao Wang & Zhang-you Chen

  3. School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu, 611731, Sichuan, People’s Republic of China

    Zhong-bao Wang

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  1. Xue Chen
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  2. Zhong-bao Wang
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  3. Zhang-you Chen
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Correspondence to Zhong-bao Wang.

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This work was supported by the National Natural Science Foundation of China (11701479, 11526170,11701478,11771067) and the Chinese Postdoctoral Science Foundation (2018M643434).

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Chen, X., Wang, Zb. & Chen, Zy. A New Method for Solving Variational Inequalities and Fixed Points Problems of Demi-Contractive Mappings in Hilbert Spaces. J Sci Comput 85, 18 (2020). https://doi.org/10.1007/s10915-020-01327-5

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  • DOI: https://doi.org/10.1007/s10915-020-01327-5

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