Skip to main content
SpringerLink
Log in

Pseudomonotone Complementarity Problems and Variational Inequalities

  • Chapter
Handbook of Generalized Convexity and Generalized Monotonicity

Part of the book series: Nonconvex Optimization and Its Applications ((NOIA,volume 76))

Abstract

In this chapter, we report recent results mainly on existence for complementarity problems and variational inequalities in infinite-dimensional spaces under generalized monotonicity, especially (algebraic) pseudomonotonicity. Variational inequalities associated to a topological pseudomonotone operator have been also considered and some possible extensions of complementarity problems and variational inequalities have been included. Finally some discussions on the equivalence of complementarity problems for pseudomonotone operators are given.

This research was supported by a grant from the National Science of the Republic of China.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 169.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 249.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 219.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Allen, G., Variational inequalities, complementarity problems and duality theorems, J. Math. Anal. Appl. 58, 1977, 1–10.

    Article  MATH  MathSciNet  Google Scholar 

  2. Aubin, J. P., Mathematical Methods of Game and Economic Theory, revised ed., Stud. Math. Appl., 7, North-Holland, Amsterdam 1982.

    Google Scholar 

  3. Baiocchi, C. and A. Capelo, Variational and Quasivariational Inequalities: Application to Free Boundary Problems, Wiley, New York 1984.

    Google Scholar 

  4. Barbu, V. and Th. Precupanu, Convexity and Optimization in Banach Spaces, Sijthoff and Nourdhoff, Bucarest, Romania 1986.

    Google Scholar 

  5. Bazarra, M. S., Goode, J. J. and M. Z. Nashed, A nonlinear complementarity problem in mathematical programming in Banach space, Proc. Am. Math. Soc. 35, 1972, 165–170.

    Google Scholar 

  6. Berge, Espace Topologiques-Functions Multivoques, Dunod, Paris 1966.

    Google Scholar 

  7. Blum, E. and W. Oettli, From Optimization and Variational Inequalities to Equilibrium Problems, The Mathematics Student 63, 1994, 123–145.

    MathSciNet  Google Scholar 

  8. Bourbaki, N., Espaces Vectoriels Topologiques, Hermann, Paris, 1966.

    Google Scholar 

  9. Borwein, J. M., Alterative theorems for general complementarity problems, Lecture Notes in Economics and Mathematical Systems Springer-Verlag, New York, 98, 1984, 328–331.

    Google Scholar 

  10. Borwein, J. M., Generalized linear complementarity problem treated without fixed point theory, J. Optimization Theory Appl. 22, 1984, 343–356.

    Google Scholar 

  11. Borwein, J. M. and S. Fitzpatrick, Local boundedness of monotone operators under minimal hypotheses, Bull. Aust. Math. Soc. 39, 1988, 439–441.

    MathSciNet  Google Scholar 

  12. Brézis, H., Equations et inequations non linéaires dans les espaces vectoriels en dualité, Ann. Inst. Fourier 18, 1968, 115–175.

    MATH  Google Scholar 

  13. Brézis, H., Nirenberg L. and G. Stampacchia, A remark on Ky Fan’s minimax principle, Boll. Unione Mat. Ital. (4)6, 1972, 293–300.

    Google Scholar 

  14. Browder, F., Nonlinear elliptic boundary value problems, Bull. Am. Math. Soc. 69, 1963, 862–874.

    MATH  MathSciNet  Google Scholar 

  15. Browder, F., Nonlinear Operators and Nonlinear Equations of Evolution in Banach Spaces, Proc. Symp. Pure Math. 18, Part 2 (the entire issue). Amer. Math. Soc., Providence, RI, 1976 (preprint version 1968).

    Google Scholar 

  16. Browder, F., Coincidence theorems, minimax theorems, and variational inequalities, Contemporary Math. 26 1984 67–80.

    MATH  MathSciNet  Google Scholar 

  17. Chadli, O., Chbani, Z. and H. Riahi, Equilibrium Problems with Generalized Monotone Bifunctions and Applications to Variational Inequalities J. Optimization Theory Appl. 105, 2000, 299–323.

    MathSciNet  Google Scholar 

  18. Chadli, O., Chbani., Z. and H. Riahi, Recession Methods for Equilibrium Problems and Applications to Variational and Hemivariational Inequalities, Discrete Contin. Dyn. Syst. 5, 1999, 185–195.

    MathSciNet  Google Scholar 

  19. Chadli, O., Chbani, Z. and H. Riahi, Equilibrium problems and noncoercive variational inequalities, Optimization 50, 2001, 17–27.

    MathSciNet  Google Scholar 

  20. Chan, D. and J. S. Pang The generalized quasi-variational inequality problem, Math. Oper. Res. 7, 1982, 211–222.

    MathSciNet  Google Scholar 

  21. Chowdhury, M. and K. K. Tan, Applications of pseudo-monotone operators with some kind of upper semicontinuity in generalized quasi-variational inequalities on non-compact sets, Proc. Am. Math. Soc. 10, 1998 2957–2968.

    MathSciNet  Google Scholar 

  22. Cottle, R. W., Notes on a fundamental theorem in quadratic programming, SIAM J. Appl. Math. 12, 1964, 663–665.

    Article  MATH  MathSciNet  Google Scholar 

  23. Cottle, R. W., Nonlinear programs with positively bounded Jacobians, SIAM J. Appl. Math. 14, 1966, 147–158.

    Article  MATH  MathSciNet  Google Scholar 

  24. Cottle, R. W., Complementarity and variational problems Sympos. Math., Instituto Nazionale di Alta Mathematica, Volume XIX, 1976, 177–208.

    MathSciNet  Google Scholar 

  25. Cottle, R. W., Numerical methods for complementarity problems in engineering and applied science, Technical Report SOL 77-24, Systems Optimization Laboratory, Department of Operations Research, Stanford University, 1977.

    Google Scholar 

  26. Cottle, R. W., Habetler, G. J. and C. E. Lemke, Quadratic forms semi-definite over convex cones, in Proceedings of the Princeton Symposium on Mathematical Programming, H. W. Kuhn, ed., Princeton University Press, Princeton, New Jersey, 1970, 551–565.

    Google Scholar 

  27. Cottle, R. W. and J. C. Yao, Pseudomonotone complementarity problems in Hilbert space, J. Optimization Theory Appl. 75(2), 1992, 281–295.

    Article  MathSciNet  Google Scholar 

  28. Cubiotti, P., General nonlinear variational inequalities with (S)12 operators, Appl. Math. Lett. 10(2), 1997 11–15.

    Article  MATH  MathSciNet  Google Scholar 

  29. Cubiotti, P. and J. C. Yao, Multivalued operators and generalized variational inequalities, Comput. Math. Appl. 29(12). 1995, 49–56.

    MathSciNet  Google Scholar 

  30. Cryer, C.W. and M. A. H. Dempster, Equivalence of linear complementarity problems and linear program in vector lattice Hilbert spaces, SIAM J. Control Optimization 18, 1980, 76–90.

    Article  MathSciNet  Google Scholar 

  31. Dantzig, G. B. and R. W. Cottle, Positive semi-definite programming, in Nonlinear Programming, Abadie, ed., A course, North-Holland, Amsterdam, 1967, 55–73.

    Google Scholar 

  32. Dash, A. T. and S. Nanda, A complementarity problem in mathematical programming in Banach space, J. Math. Anal. Appl. 98, 1984, 328–331.

    Article  MathSciNet  Google Scholar 

  33. Deimling, K., Nonlinear Functional Analysis, Springer-Verlag, Berlin, 1985.

    Google Scholar 

  34. Do, C., Bifurcation theory for elastic plates subjected to unilateral conditions, J. Math. Anal. Appl. 60, 1977, 435–448.

    Article  MATH  MathSciNet  Google Scholar 

  35. Dorn, W. S., Self-dual quadratic programs, SIAM J. Appl. Math. 9(1), 1961, 51–54.

    Article  MATH  MathSciNet  Google Scholar 

  36. Ekeland, I. and R. Temam, Convex Analysis and Variational Problems, North-Holland, Amsterdam, 1976.

    Google Scholar 

  37. Fan, K., A generalization of Tychonoff’s fixed point theorem, Math. Ann. 142, 1961, 305–310.

    Article  MATH  MathSciNet  Google Scholar 

  38. Fang, S.C., A linearization method for generalized complementarity problems, IEEE Trans. Autom. Control 38, 1982, 363–383.

    MATH  Google Scholar 

  39. Fang, S. C. and E. L. Peterson, Generalized variational inequalities, J. Optimization Theory Appl. 38, 1982, 363–383.

    Article  MathSciNet  Google Scholar 

  40. Guo, J. S. and J. C. Yao, Extension of strongly nonlinear quasivariational inequalities, Appl. Math. Lett. 5(3), 1992, 35–38.

    MathSciNet  Google Scholar 

  41. Guo, J. S. and J. C. Yao, Variational inequalities with nonmonotone operators, J. Optimization Theory Appl. 80(1), 1994, 63–74.

    Article  MathSciNet  Google Scholar 

  42. Habetler, G. J. and A. J. Price, Existence theory for generalized nonlinear complementarity problems, J. Optimization Theory Appl. 7, 1971, 223–239.

    Article  MathSciNet  Google Scholar 

  43. Hadjisavvas, N. and S. Schaible, Quasimonotone variational inequalities in Banach spaces, J. Optimization Theory Appl. 90, 1996, 95–111.

    Article  MathSciNet  Google Scholar 

  44. Hadjisavvas, N. and S. Schaible, On strong pseudomonotonicity and (semi)strict quasimonotonicity, J. Optimization Theory Appl. 79, 1993, 139–155.

    Article  MathSciNet  Google Scholar 

  45. Harker, P. T. and J. S. Pang, Finite-dimensional variational inequality and nonlinear complementarity problems: A survey of theory, algorithms and applications, Math. Program., Series B, 48, 1990, 161–220.

    MathSciNet  Google Scholar 

  46. Hartmann, P. and G. Stampacchia, On some nonlinear elliptic differential functional equations, Acta Math. 115, 1966, 153–188.

    Google Scholar 

  47. Holmes, R. B., Geometric Functional Analysis and Its Applications, Springer-Verlag, New York, 1975.

    Google Scholar 

  48. Ingleton, A. W., A problem in linear inequalities, Proc. London Math. Soc. 16, 1966, 519–536.

    MATH  MathSciNet  Google Scholar 

  49. Isac, G., Complementarity Problems, Lecture Notes in Mathematics 1528, Springer-Verlag, Berlin Heidelberg, 1992.

    Google Scholar 

  50. Isac, G., Nonlinear complementarity problem and Galerkin method, J. Math. Anal. Appl. 108, 1985, 563–574.

    Article  MathSciNet  Google Scholar 

  51. Isac, G. and M. S. Gowda, Operators of class (S)1+, Altman’s condition and the complementarity problem, J. Fac. Sci., Univ. Tokyo, Sect. IA, Math. 40, 1993, 1–16.

    MathSciNet  Google Scholar 

  52. Isac, G. and M. Théra, Complementarity problems and the existence of the post-critical equilibrium state of a thing elastic plate, J. Optimization Theory Appl. 58, 1988, 241–257.

    Article  Google Scholar 

  53. Jou, C. R. and J. C. Yao, Extension of generalized variational inequalities, Appl. Math. Lett. 6, 1993, 21–25.

    Article  MathSciNet  Google Scholar 

  54. Karamardian, S., Duality in Mathematical Programming, P.h.D. Thesis, University of California, Berkley, 1966.

    Google Scholar 

  55. Karamardian, S., Generalized complementarity problems, J. Optimization Theory Appl. 8, 1971, 161–168.

    Article  MATH  MathSciNet  Google Scholar 

  56. Karamardian, S., Complementarity problems over cones with monotone and pseudomonotone maps, J. Optimization Theory Appl. 18, 1976, 445–454.

    Article  MATH  MathSciNet  Google Scholar 

  57. Karamardian, S., An existence theorem for the complementarity problem, J. Optimization Theory Appl. 19(2), 1976, 227–232.

    Article  MATH  MathSciNet  Google Scholar 

  58. Karamardian, S. and S. Schaible, Seven kinds of monotone maps, J. Optimization Theory Appl. 66, 1990, 37–46.

    Article  MathSciNet  Google Scholar 

  59. Kelley, J. L. and I. Namioka, Linear Topolical Spaces, Springer, New York, 1976.

    Google Scholar 

  60. Kinderlehrer, D. and G. Stampacchia, An Introduction to Variational Inequalities and Their Applications, Academic Press, New york, 1980.

    Google Scholar 

  61. Kluge, R., Nichtlineare Variationsungleichungen und Extremalaufgaben. Verl. der Wiss., Berlin, 1979.

    Google Scholar 

  62. Knaster, B., Kuratowski, C. and S. Mazukiewicz, Ein bewies des fixpunktsatzes für simplexe, Fund. Math. 14, 1929, 132–137.

    Google Scholar 

  63. Kneser, H., Sur un Théorème fondamental de la théorie des jeux, C. R. Acad. Sci. Paris Sr. I Math. 234, 1952, 2418–2420.

    MathSciNet  Google Scholar 

  64. Lemke, C. E., Bimatrix equilibrium points and mathematical programming, Managment Sciences 11(7), 1965, 681–689.

    MATH  MathSciNet  Google Scholar 

  65. Lions, J. L. and G. Stampacchia, Variational inequalities, Comm. Pure Appl. Math. 20, 1967, 493–519.

    MathSciNet  Google Scholar 

  66. Luna, G., A remark on the nonlinear complementarity problem, Proc. Amer. Math. Soc. 48, 1975, 132–134.

    MATH  MathSciNet  Google Scholar 

  67. Mancino, O. and G. Stampacchia, Convex programming and variational inequalities, J. Optimization Theory Appl. 9, 1972, 3–23.

    Article  MathSciNet  Google Scholar 

  68. Minty, G. J., Monotone (non-linear) operators in Hilbert spaces, Duke Math. J. 29, 1962, 341–346.

    Article  MATH  MathSciNet  Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

  70. Megiddo, N. and M. Kojima, On the existence and uniqueness of solutions in nonlinear complementarity theory, Math. Program. 12, 1977, 110–130.

    MathSciNet  Google Scholar 

  71. Moré, J. J. and W. C. Rheinboldt, On P-and S-functions and related classes of nonlinear mappings, Linear Algebra Appl. 6, 1973, 45–68.

    Google Scholar 

  72. Mosco, U., A nonlinear complementarity problem in Banach spaces, J. Math. Anal. Appl. 21, 1980, 351–356.

    Google Scholar 

  73. Mosco, U., Implicit Variational Problems and Quasi-Variational Inequalities, Lecture Notes in Mathematics, 543, Springer, Berlin, 1976.

    Google Scholar 

  74. Noor, M. A., The quasi-complementarity problem, J. Math. Anal. Appl. 130, 1988, 344–353.

    Article  MATH  MathSciNet  Google Scholar 

  75. Parida, J. and A. Sen, A variational-like inequality for multifuctions with applications, J. Math. Anal. Appl. 924, 1987, 73–81.

    MathSciNet  Google Scholar 

  76. Phelps, R. R., Convex Functions, Monotone Operators and Differentiability. Lecture Notes in Mathematics, 1364, Springer-Verlag, 1993.

    Google Scholar 

  77. Rheinboldt, W. C., On M-functions and their application to nonlinear Gauss-Seidel iterations and network flows, J. Math. Anal. Appl. 32, 1970, 274–307.

    Article  MATH  MathSciNet  Google Scholar 

  78. Riddel, R. C., Equivalence of nonlinear complementarity problems and least element problems in Banach lattices, Math. Oper. Res. 6, 1981, 462–474.

    MathSciNet  Google Scholar 

  79. Saigal, R., Extension of the generalized complementarity problem, Math. Oper. Res. 1, 1976, 260–266.

    MATH  MathSciNet  Google Scholar 

  80. Schaible, S. and J. C. Yao, On the equivalence of nonlinear complementarity problems and least-element problems, Math. Program. 70, 1995, 191–200.

    Article  MathSciNet  Google Scholar 

  81. Shih, M. H. and K. K. Tan, Browder-Hartmann-Stampacchia variational inequalities for multi-valued monotone operators, J. Math. Anal. Appl. 134, 1988, 431–440

    Article  MathSciNet  Google Scholar 

  82. Stampacchia, G., Variational inequalities, in Theory and Applications of Monotone Operators, A. Ghizzetti ed., Proceedings of the NATO Advanced Study Institute, Venice Italys (Edizioni Oderisi, Gubbio, Italy), 1968, 102–192.

    Google Scholar 

  83. Théra, M., Existence results for the nonlinear complementarity problem and applications to nonlinear analysis, J. Math. Anal. Appl. 154, 1991, 572–584.

    MATH  MathSciNet  Google Scholar 

  84. Théra, M., A note on the Hartmann-Stampacchia thoerem, in Nonlinear Analysis and Applications, V. Lakshmikanthan ed., Dekker, New York, 1987, 573–577.

    Google Scholar 

  85. Watson, P. J., Variational inequalities in nonreflexive Banach spaces, Appl. Math. Lett. 10(2), 1997, 45–48.

    Article  MATH  MathSciNet  Google Scholar 

  86. Watson, L. M., Solving the nonlinear complementarity problem by a homotopy method, SIAM J. Control Optim. 17, 1979, 36–46.

    Article  MATH  MathSciNet  Google Scholar 

  87. Yao, J. C., Variational inequalities with generalized monotone operators, Math. Oper. Res. 19(3), 1994, 691–705.

    MATH  MathSciNet  Google Scholar 

  88. Yao, J. C., On the generalized complementarity problem, J. Austral. Math. Soc. Ser. B 35, 1994, 420–428.

    MATH  MathSciNet  Google Scholar 

  89. Yao, J. C., Existence of generalized Variational inequalities, Oper. Res. Lett. 15, 1994, 35–40.

    Article  MATH  MathSciNet  Google Scholar 

  90. Yao, J. C., Multi-valued variational inequalities with Kpseudomonotone operators, J. Optimization Theory Appl. 83(2), 1994, 391–403.

    Article  MATH  Google Scholar 

  91. Yao, J. C., Variational inequality, Appl. Math. Lett. 5(1), 1992, 39–42.

    MATH  MathSciNet  Google Scholar 

  92. Yao, J. C., The generalized quasi-variational inequality problem with applications, J. Math. Anal. Appl. 158(1), 1991, 139–160.

    MATH  MathSciNet  Google Scholar 

  93. Zeidler, E., Nonlinear Functional Analysis and its Applications, Vol II/B, Nonlinear Monotone Operators, Springer-Verlag, 1985.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Applied Mathematics, National Sun Yat-Sen University, Kaohsiung, 80424, Taiwan, ROC

    Jen-Chih Yao

  2. Department of Economics, Faculty of Economics and Social Sciences, University Ibn Zohr, Agadir, B.P. 8658, Hay Dakhla, Agadir, Morocco

    Ouayl Chadli

Authors
  1. Jen-Chih Yao
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ouayl Chadli
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. University of Aegean, Greece

    Nicolas Hadjisavvas

  2. University of Pécs, Hungary

    Sándor Komlósi

  3. University of California, Riverside, USA

    Siegfried Schaible

Rights and permissions

Reprints and permissions

Copyright information

© 2005 Springer Science + Business Media, Inc.

About this chapter

Cite this chapter

Yao, JC., Chadli, O. (2005). Pseudomonotone Complementarity Problems and Variational Inequalities. In: Hadjisavvas, N., Komlósi, S., Schaible, S. (eds) Handbook of Generalized Convexity and Generalized Monotonicity. Nonconvex Optimization and Its Applications, vol 76. Springer, New York, NY. https://doi.org/10.1007/0-387-23393-8_12

Download citation

Publish with us

Policies and ethics

Search

Navigation