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Explicit Hierarchical Fixed Point Approach to Variational Inequalities

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Abstract

An explicit hierarchical fixed point algorithm is introduced to solve monotone variational inequalities, which are governed by a pair of nonexpansive mappings, one of which is used to define the governing operator and the other to define the feasible set. These kinds of variational inequalities include monotone inclusions and convex optimization problems to be solved over the fixed point sets of nonexpansive mappings. Strong convergence of the algorithm is proved under different circumstances of parameter selections. Applications in hierarchical minimization problems are also included.

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References

  1. Baiocchi, C., Capelo, A.: Variational and Quasi-Variational Inequalities. Wiley, New York (1984)

    Google Scholar 

  2. Cottle, R.W., Giannessi, F., Lions, J.L.: Variational Inequalities and Complementarity Problems: Theory and Applications. Wiley, New York (1980)

    MATH  Google Scholar 

  3. Facchinei, F., Pang, J.-S.: Finite-Dimensional Variational Inequalities and Complementarity Problems, vols. I & II. Springer, Berlin (2003)

    Google Scholar 

  4. Giannessi, F., Maugeri, A.: Variational Inequalities and Network Equilibrium Problems. Plenum, New York (1995)

    MATH  Google Scholar 

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

    MATH  Google Scholar 

  6. Auslender, A., Teboulle, M.: Interior projection-like methods for monotone variational inequalities. Math. Program. 104, 39–68 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  7. He, B.S.: A class of projection and contraction methods for monotone variational inequalities. Appl. Math. Optim. 35, 69–76 (1997)

    MathSciNet  MATH  Google Scholar 

  8. He, B.S.: Inexact implicit methods for monotone general variational inequalities. Math. Program. 86, 113–123 (1999)

    Article  Google Scholar 

  9. He, B.S., Li, M., Liao, L.-Z.: An improved contraction method for structured monotone variational inequalities. Optimization 57, 643–653 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  10. He, B.S., Xu, M.-H.: A general framework of contraction methods for monotone variational inequalities. Pac. J. Optim. 4, 195–212 (2008)

    MathSciNet  MATH  Google Scholar 

  11. He, B., He, X.Z., Liu, H.X., Wu, T.: Self-adaptive projection method for co-coercive variational inequalities. Eur. J. Oper. Res. 196, 43–48 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  12. He, B., Wang, X., Yang, J.: A comparison of different contraction methods for monotone variational inequalities. J. Comput. Math. 27, 459–473 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  13. He, B.S., Yang, Z.H., Yuan, X.M.: An approximate proximal-extragradient type method for monotone variational inequalities. J. Math. Anal. Appl. 300, 362–374 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  14. Wang, Y., Xiu, N., Wang, C.: A new version of extragradient method for variational inequality problems. Comput. Math. Appl. 42, 969–979 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  15. Xiu, N., Wang, C., Zhang, J.: Convergence properties of projection and contraction methods for variational inequality problems. Appl. Math. Optim. 43, 147–168 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  16. Yang, X.Q.: Vector variational inequality and its duality. Nonlinear Anal. 21, 869–877 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  17. Giannessi, F., Mastroeni, G., Yang, X.Q.: A survey on vector variational inequalities. Boll. Unione Mat. Ital. (9) 2(1), 225–237 (2009)

    MathSciNet  Google Scholar 

  18. Marino, G., Xu, H.K.: A general iterative method for nonexpansive mappings in Hilbert spaces. J. Math. Anal. Appl. 318, 43–52 (2006)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  21. Xu, H.K.: Viscosity approximation methods for nonexpansive mappings. J. Math. Anal. Appl. 298, 279–291 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  22. Yamada, I.: The hybrid steepest descent for the variational inequality problems over the intersection of fixed point sets of nonexpansive mappings. In: Inherently Parallel Algorithms in Feasibility and Optimization and Their Applications, pp. 473–504. Elservier, New York (2001)

    Google Scholar 

  23. Mainge, P.-E., Moudafi, A.: Strong convergence of an iterative method for hierarchical fixed-points problems. Pac. J. Optim. 3, 529–538 (2007)

    MathSciNet  MATH  Google Scholar 

  24. Moudafi, A., Mainge, P.-E.: Towards viscosity approximations of hierarchical fixed-points problems. Fixed Point Theory Appl. 2006, 1–10 (2006). Article ID 95453

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  26. Ceng, L.C., Ansari, Q.H., Yao, J.C.: Mann type steepest-descent and modified hybrid steepest-descent methods for variational inequalities in Banach spaces. Numer. Funct. Anal. Optim. 29, 987–1033 (2008)

    Article  MathSciNet  Google Scholar 

  27. Ceng, L.C., Ansari, Q.H., Yao, J.C.: On relaxed viscosity iterative methods for variational inequalities in Banach spaces. J. Comput. Appl. Math. 230, 813–822 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  28. Ceng, L.C., Xu, H.K., Yao, J.C.: A hybrid steepest-descent method for variational inequalities in Hilbert spaces. Appl. Anal. 87, 575–589 (2008)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  30. Lions, P.L.: Approximation de points fixes de contractions. C.R. Acad. Sci. Sèr. A–B Paris 284, 1357–1359 (1977)

    MATH  Google Scholar 

  31. Reich, S.: Strong convergence theorems for resolvents of accretive operators in Banach spaces. J. Math. Anal. Appl. 75, 287–292 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  32. Reich, S.: Approximating fixed points of nonexpansive mappings, Panamerican. Math. J. 4, 23–28 (1994)

    MATH  Google Scholar 

  33. Shioji, N., Takahashi, W.: Strong convergence of approximated sequences for nonexpansive mappings in Banach spaces. Proc. Am. Math. Soc. 125, 3641–3645 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  34. Tam, N.N., Yao, J.C., Yen, N.D.: On some solution methods for pseudomonotone variational inequalities. J. Optim. Theory Appl. 138, 253–273 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  35. Wittmann, R.: Approximation of fixed points of nonexpansive mappings. Arch. Math. 58, 486–491 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  36. Wong, N.C., Sahu, D.R., Yao, J.C.: Solving variational inequalities involving nonexpansive type mappings. Nonlinear Anal., Theory Methods Appl. 69, 4732–4753 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  37. Xu, H.K.: Another control condition in an iterative method for nonexpansive mappings. Bull. Aust. Math. Soc. 65, 109–113 (2002)

    Article  MATH  Google Scholar 

  38. Xu, H.K.: Remarks on an iterative method for nonexpansive mappings. Commun. Appl. Nonlinear Anal. 10, 67–75 (2003)

    MATH  Google Scholar 

  39. Xu, H.K.: An iterative approach to quadratic optimization. J. Optim. Theory Appl. 116, 659–678 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  40. Yao, J.C., Chadli, O.: Pseudomonotone complementarity problems and variational inequalities. In: Crouzeix, J.P., Haddjissas, N., Schaible, S. (eds.) Handbook of Generalized Convexity and Monotonicity, pp. 501–558 (2005)

    Chapter  Google Scholar 

  41. Zeng, L.C., Wong, N.C., Yao, J.C.: On the convergence analysis of modified hybrid steepest-descent methods with variable parameters for variational inequalities. J. Optim. Theory Appl. 132, 51–69 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  42. Xu, H.K.: Viscosity method for hierarchical fixed Point approach to variational inequalities. Taiwan. J. Math. 14, 463–478 (2010)

    MATH  Google Scholar 

  43. Geobel, K., Kirk, W.A.: Topics in Metric Fixed Point Theory. Cambridge Studies in Advanced Mathematics, vol. 28. Cambridge University Press, Cambridge (1990)

    Book  Google Scholar 

  44. Attouch, H.: Variational Convergence for Functions and Operators. Applicable Math. Series. Pitman, London (1984)

    Google Scholar 

  45. Baillon, J.B., Haddad, G.: Quelques proprietes des operateurs angle-bornes et n-cycliquement monotones. Isr. J. Math. 26, 137–150 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  46. Cabot, A.: Proximal point algorithm controlled by a slowly vanishing term: applications to hierarchical minimization. SIAM J. Optim. 15, 555–572 (2005)

    Article  MathSciNet  MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. Dipartimento di Matematica, Universita della Calabria, 87036, Arcavacata di Rende, Cs, Italy

    Giuseppe Marino

  2. Department of Applied Mathematics, National Sun Yat-sen University, Kaohsiung, 804, Taiwan

    Hong-Kun Xu

  3. Department of Mathematics, College of Science, King Saud University, PO Box 2455, Riyadh, 11451, Saudi Arabia

    Hong-Kun Xu

Authors
  1. Giuseppe Marino
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  2. Hong-Kun Xu
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Corresponding author

Correspondence to Hong-Kun Xu.

Additional information

Communicated by X.-Q. Yang.

Research of H.-K. Xu was supported in part by NSC 97-2628-M-110-003-MY3.

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Marino, G., Xu, HK. Explicit Hierarchical Fixed Point Approach to Variational Inequalities. J Optim Theory Appl 149, 61–78 (2011). https://doi.org/10.1007/s10957-010-9775-1

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