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A New Double-Projection Method for Solving Variational Inequalities in Banach Spaces

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Abstract

In this paper, we study the variational inequalities involving monotone and Lipschitz continuous mapping in Banach spaces. A new and simple iterative method, which combines Halpern’s technique and the subgradient extragradient idea, is given. Under mild and standard assumptions, we establish the strong convergence of our algorithm in a uniformly smooth and convex Banach spaces. We also present a modification of our method using a line-search approach, this enable to obtain strong convergence in real and reflexive Banach spaces, without the prior knowledge of the Lipschitz constant. Numerical experiments illustrate the performances of our new algorithm and provide a comparison with related algorithms. Our results generalize and extend some of the existing works in Hilbert spaces to Banach spaces as well as provide an extension from weak to strong convergence.

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References

  1. 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 

  2. 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 

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

    MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

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

    MATH  Google Scholar 

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

    MATH  Google Scholar 

  7. Glowinski, R., Lions, J.-L., Trémolières, R.: Numerical Analysis of Variational Inequalities. Elsevier, Amsterdam (1981)

    MATH  Google Scholar 

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

    MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

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

    MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

  13. 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 

  14. 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 

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

    Article  MathSciNet  MATH  Google Scholar 

  16. 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 

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

    MathSciNet  Google Scholar 

  18. Browder, F.E.: Convergence of approximants to fixed points of nonexpansive nonlinear mappings in Banach spaces. Arch. Ration. Mech. Anal. 24, 82–90 (1967)

    Article  MATH  Google Scholar 

  19. Halpern, B.: Fixed points of nonexpansive maps. Bull. Am. Math. Soc. 73, 957–961 (1967)

    Article  MATH  Google Scholar 

  20. 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)

    Article  MathSciNet  MATH  Google Scholar 

  21. Iusem, A.N., Nasri, M.: Korpelevich’s method for variational inequality problems in Banach spaces. J. Glob. Optim. 50, 59–76 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  22. Beauzamy, B.: Introduction to Banach Spaces and Their Geometry. North-Holland, Amsterdam (1985)

    MATH  Google Scholar 

  23. Ball, K., Carlen, E.A., Lieb, E.H.: Sharp uniform convexity and smoothness inequalities for trace norms. Invent. Math. 115, 463–482 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  24. Minty, G.J.: On a monotonicity method for the solution of non-linear equations in Banach spaces. Proc. Natl. Acad. Sci. USA 50, 1038–1041 (1963)

    Article  MathSciNet  MATH  Google Scholar 

  25. Alber, Y.I.: Metric and generalized projection operator in Banach spaces: properties and applications. Theory and Applications of Nonlinear Operators of Accretive and Monotone Type, 178 of Lecture Notes in Pure and Applied Mathematics, pp. 15–50. Dekker, New York (1996)

    Google Scholar 

  26. Aoyama, K., Kohsaka, F.: Strongly relatively nonexpansive sequences generated by firmly nonexpansive-like mappings. Fixed Point Theory Appl 2014, 95 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  27. Maingé, P.-E.: Strong convergence of projected subgradient methods for nonsmooth and nonstrictly convex minimization. Set-Valued Anal. 16, 899–912 (2008)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  29. Browder, F.: Nonlinear monotone operators and convex sets in Banach spaces. Bull. Am. Math. Soc. 71, 780–785 (1965)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  31. Takahashi, W.: Nonlinear Functional Analysis. Yokohama Publishers, Yokohama (2000)

    MATH  Google Scholar 

  32. Moudafi, A.: Viscosity methods for fixed points problems. J. Math. Anal. Appl. 241, 46–55 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  33. Takahashi, W., Takeuchi, Y., Kubota, R.: Strong convergence theorems by hybrid methods for families of nonexpansive mappings in Hilbert spaces. J. Math. Anal. Appl. 341, 276–286 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  34. Haugazeau, Y.: Sur les Inéquations Variationnelles et la Minimisation de Fonctionnelles Convexes. Thèse. Université de Paris, Paris (1968)

    Google Scholar 

  35. Bauschke, H.H., Combettes, P.L., Luke, D.R.: A strongly convergent reflection method for finding the projection onto the intersection of two closed convex sets in a Hilbert space. J. Approx. Theory 141, 63–69 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  36. Bauschke, H.H., Borwein, J.M., Combettes, P.L.: Essential smoothness, essential strict convexity, and Legendre functions in Banach spaces. Comm. Contemp. Math. 3, 615–647 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  37. Hieu, D.V.: Common solutions to pseudomonotone equilibrium problems. Bull. Iran. Math. Soc. 42, 1207–1219 (2016)

    MathSciNet  MATH  Google Scholar 

  38. Hieu, D.V., Anh, P.K., Muu, L.D.: Modified hybrid projection methods for finding common solutions to variational inequality problems. Comput. Optim. Appl. 66, 75–96 (2017)

    Article  MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

  40. Harker, P.T., Pang, J.-S.: A damped-Newton method for the linear complementarity problem. In: Allgower, G., Georg, K. (eds.) Lectures in Applied Mathematics Computational Solution of Nonlinear Systems of Equations, vol. 26, pp. 265–284. AMS, Providence (1990)

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  42. 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 

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Acknowledgements

We are very grateful to the anonymous referee and Associate Editor whose insightful comments helped to considerably improve an earlier version of this paper. The first author is supported by the NSF of China (Grant No. 11771063), the Natural Science Foundation of Chongqing (cstc2017jcyjAX0006), Science and Technology Project of Chongqing Education Committee (Grant No. KJ KJ1703041), the University Young Core Teacher Foundation of Chongqing (020603011714), Talent Project of Chongqing Normal University (Grant No. 02030307-00024). The second author work is supported by the EU FP7 IRSES program STREVCOMS, Grant No. PIRSES-GA-2013-612669. The fourth author is grateful to the Alexander von Humboldt Foundation and the Institute of Mathematics of the University of Würzburg, Germany for their support and hospitality.

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Authors and Affiliations

  1. School of Mathematics Science, Chongqing Normal University, Chongqing, 401331, China

    Gang Cai

  2. Department of Mathematics, ORT Braude College, 2161002, Karmiel, Israel

    Aviv Gibali

  3. Department of Mathematical Sciences, University of Wisconsin-Milwaukee, Milwaukee, WI, USA

    Olaniyi S. Iyiola

  4. Department of Mathematics, University of Nigeria, Nsukka, Nigeria

    Yekini Shehu

Authors
  1. Gang Cai
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  2. Aviv Gibali
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  3. Olaniyi S. Iyiola
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  4. Yekini Shehu
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Correspondence to Aviv Gibali.

Additional information

Communicated by Alfredo N. Iusem.

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Cai, G., Gibali, A., Iyiola, O.S. et al. A New Double-Projection Method for Solving Variational Inequalities in Banach Spaces. J Optim Theory Appl 178, 219–239 (2018). https://doi.org/10.1007/s10957-018-1228-2

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