Pseudomonotone and Implicit Complementarity Problems

  • G. Isac
  • V. A. Bulavsky
  • V. V. Kalashnikov
Part of the Nonconvex Optimization and Its Applications book series (NOIA, volume 63)


In this chapter, we extend our techniques to the case of infinite qimensional complementarity problems. Especial attention is paid to the latter with pseu-domonotone operators. The second part of the chapter is devoted to Implicit Complementarity Problems with single-valued and multi-valued mappings.


Complementarity Problem Convex Cone Multivalued Mapping Nonempty Closed Convex Subset Nonlinear Complementarity Problem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Aubin, J.-P. Nonlinear Analysis and Its Economic Applications. Moscow: Mir, 1988 (translation into Russian). Google Scholar
  2. Avtukhovich, EV. “Existence of solution for multi-valued complementarity problem”, Working Paper, Computer Center of the Russian Academy of Sciences, Moscow, 1999 (in Russian). Google Scholar
  3. Baiocchi, C, and Capelo, A. Variational Inequalities and Quasi-Variational Inequalities: Applications to Free Boundary Value Problems. New York: Wiley and Sons, 1984.Google Scholar
  4. Bakushinskii, A.B., and Goncharskii, A.V. Iterative Methods for Ill-Posed Problems. Dordrecht: Kluwer Academic Publishers, 1994.Google Scholar
  5. Ben-el-Mechaiekh H, Idzik A. A Leray-Schauder type theorem for approximable maps. Proc. Amer. Math. Soc. 1994; 122:105–109.MathSciNetMATHCrossRefGoogle Scholar
  6. Ben-el-Mechaiekh H, Chebbi S, Florenzano M. A Leray-Schauder type theorem for approximable maps: a simple proof. Proc.Amer. Math. Soc. 1998;126:2345–2349.MathSciNetMATHCrossRefGoogle Scholar
  7. Bensoussan A, Lions J-L. Nouvelle formulation de problemes de controle impulsionnel et applications. C.R.Acad.Sci., Paris, 1973; Ser.A-B 276:1189–1192.MathSciNetMATHGoogle Scholar
  8. Bensoussan A, Gourset M., Lions J-L. Controle impulsionnel et inequations quasi-variationnelles stationaires. C.R.Acad.Sci., Paris, 1973; Ser. A-B 276:1279–1284.MATHGoogle Scholar
  9. Browder FE. Existence and approximation of solution of nonlinear variational inequalities. Proc. National Acad. Sci., USA, 1966; 56:1080–1086.MathSciNetMATHCrossRefGoogle Scholar
  10. Browder FE, and Ton BA. Nonlinear functional equations in Banach spaces and elliptic super regularization. Math. Zeitschr. 1968; 105:177–195.MathSciNetMATHCrossRefGoogle Scholar
  11. Bulavsky, VA, Isac, G, Kalashnikov, VV. 1. ‘Application of topological degree theory to complementarity problems’.- In: Multilevel Optimization: Algorithms and Applications, A. Migdalas, P.M. Pardalos and P. Varbrand, eds. Dordrecht-Boston-London: Kluwer Academic Publishers, 1998.Google Scholar
  12. Bulavsky, VA, Isac, G, Kalashnikov, VV. 2. Application of topological degree theory to semidefinite complementarity problems. Proceedings of the International Conference on Operations Research OR’98; 1998 August 31–September 03; Zurich- Berlin-Heidelberg: Springer-Verlag, 1999.Google Scholar
  13. Bulavsky VA, Kalashnikov VV. 1. One-parametric drive method to study equilibrium. Ekonomika i Matematicheskie Metody (in Russian) 1994; 30:129–138.Google Scholar
  14. Bulavsky VA, Kalashnikov VV. 2. Equilibria in generalized Cournot and Stackelberg models. Ekonomika i Matematicheskie Metody (in Russian) 1995; 31:164–176.Google Scholar
  15. Bulavsky VA, Kalashnikov VV. 3. ‘An alternative model of spatial competition’.- In: Operations research and decision aid methodologies in traffic and transportation management, M. Labbe, G. Laporte, K. Tanczos, P. Ibint, eds. NATO ASI Series, Series F: Computer and Systems Science, Vol. 166, Berlin-Heidelberg: Springer-Verlag, 1998.Google Scholar
  16. Capuzzo-Dolcetta, J. and Mosco, U. ‘Implicit complementarity problems and quasi-variational inequalities’. In: Variational Inequalities and Complementarity Problems, Theory and Applica-tions, R.W. Cottle, F. Giannessi and J.L. Lions, eds. New York-London: John Wiley and Sons, 1980.Google Scholar
  17. Chan D, Pang J-S. The generalized quasi-variational problem. Math. Oper. Res. 1982; 7:211–222.MathSciNetMATHCrossRefGoogle Scholar
  18. Cottle, R.W., Pang, J.S., and Stone, R.E.. The Linear Complementarity Problem. New York: Academic Press, 1992.MATHGoogle Scholar
  19. Cottle RW, Yao JC. Pseudomonotone complementarity problems in Hilbert space. J. Optim. Theory Appl. 1992; 75:281–295.MathSciNetMATHCrossRefGoogle Scholar
  20. Dugundji, J., Granas, A. Fixed Point Theory. Warszawa: PWN — Polish Scientific Publishers, 1982.MATHGoogle Scholar
  21. Eaves BC. On the basic theorem of complementarity. Math. Programming 1971; 1:68–75.MathSciNetMATHCrossRefGoogle Scholar
  22. Granas A. Sur la methode de continuity de Poincare. C.R. Acad.Sci., Paris, France, 1976; 282:978–985.MathSciNetGoogle Scholar
  23. Hadjisavvas N, Schaible S. 1. On strong pseudomonotonicity, (semi)strict quasimonotonicity. J. Optim. Theory Appl. 1993; 79:139–156MathSciNetMATHCrossRefGoogle Scholar
  24. Hadjisavvas N, Schaible S. 2. Quasimonotone variational inequalities in Banach spaces. J. Optim. Theory Appl. 1996; 90:95–111.MathSciNetMATHCrossRefGoogle Scholar
  25. Harker PT, Pang JS. Finite-dimensional variational inequalities and nonlinear complementarity problems: a survey of theory, algorithms and applications. Math. Programming 1990; 48: 161–220.MathSciNetMATHCrossRefGoogle Scholar
  26. Harker PT, Choi SC. A penalty function approach for mathematical programs with variational inequality constraints. Information and Decision Technologies 1991; 17:41–50.MathSciNetMATHGoogle Scholar
  27. Hyers, D.H., Isac, G., and Rassias, T.M. Topics in Nonlinear Analysis and Applications. Singapore: World Scientific, 1997.MATHCrossRefGoogle Scholar
  28. Isac G. 1. On the implicit complementarity problem in Hilbert spaces. Bull. Austral. Math.Soc. 1985;32:251–260.MathSciNetCrossRefGoogle Scholar
  29. Isac G. 2. Complementarity problem and coincidence equations on convex cones. Boll.Un.Mat.Ital. 1986; B(6) 5:925–943.MathSciNetGoogle Scholar
  30. Isac G. 3. Fixed point theory and complementarity problems in Hilbert spaces. Bull. Austral. Math. Soc. 1987;36:295–310.MathSciNetCrossRefGoogle Scholar
  31. Isac G. 4. Fixed point theory, coincidence equations on convex cones and complementarity problem. Contemporary Mathematics 1988; 72:139–155.MathSciNetCrossRefGoogle Scholar
  32. Isac G. 5. A special variational inequality and the implicit complementarity problem. J. Fac. Sci. Univ. Tbkyo, Sect.IA, Math. 1990;37:109–127.MathSciNetMATHGoogle Scholar
  33. Isac G. 6. Complementarity Problems, Lecture Notes in Mathematics. Berlin-Heidelberg: Springer Verlag, Vol. 1528, 1992.MATHGoogle Scholar
  34. Isac G. 7. Tikhonov regularization and the complementarity problem in Hilbert spaces. J. Math. Analysis Appl. 1993; 174:53–66.MathSciNetMATHCrossRefGoogle Scholar
  35. Isac G. 8. Exceptional families of elements for fc-fields in Hilbert spaces and complementarity theory. Proceedings of the International Conference on Optimization Techniques and Applications (ICOTA’98); 1998 July 1–3; Perth, Australia.Google Scholar
  36. Isac G. 9. A generalization of Karamardian’s condition in complementarity theory. Nonlinear Analysis Forum 1999; 4:49–63.MathSciNetMATHGoogle Scholar
  37. Isac G. 10. On the solvability of multivalued complementarity problem: A topological method. Proceedings of the Fourth European Workshop on Fuzzy Decision Analysis and Recognition Technology (EFDAN’99); 1999 June 14–15; Dortmund, Germany (R. Felix, ed.), 1999.Google Scholar
  38. Isac G. 11. Exceptional family of elements, feasibility and complementarity. J. Optim.Theory Appl. 2000;104:577–588.MathSciNetMATHCrossRefGoogle Scholar
  39. Isac G. 12. ‘Exceptional family of elements, feasibility, solvability and continuous paths of e-solutions for nonlinear complementarity problems’. -In: Approximation and Complexity in Numerical Op-timization: Continuous and Discrete Problems, P.M. Pardalos, ed., 2000,Google Scholar
  40. Isac G. 13. Topological Methods in Complementarity Theory, Dordrecht: Kluwer Academic Pubishers, 2000.MATHGoogle Scholar
  41. Isac G, Bulavsky VA, Kalashnikov VV. Exceptional families, topological degree and complementarity problems. J. Global Opt. 1997; 10:207–225.MATHCrossRefGoogle Scholar
  42. Isac G, Carbone A. Exceptional families of elements for continuous functions. Some applications to complementarity theory. J. Global Opt. 1999;15:181–196.MathSciNetMATHCrossRefGoogle Scholar
  43. Isac G, Goeleven D. 1. Existence theorems for the implicit complementarity problems. Intern. J. Math, and Math. Sci. 1993; 16:67–74.MathSciNetMATHCrossRefGoogle Scholar
  44. Isac G, Goeleven D. 2. The implicit general order complementarity problem: models and iterative methods. Ann. Oper. Res. 1993; 44:63–92.MathSciNetMATHCrossRefGoogle Scholar
  45. Isac G, Kalashnikov VV. Exceptional families of elements, Leray- Schauder alternatives, pseu-domonotone operators and complementarity. J. Optim. Theory Appl. 2001; 109: 69–83.MathSciNetMATHCrossRefGoogle Scholar
  46. Isac G, Obuchowska WT. Functions without exceptional families of elements and complementarity problems. J.Optim.Theory Appl. 1998; 99:147–163.MathSciNetMATHCrossRefGoogle Scholar
  47. Isac, G., and Zhao, YB. 1. “Exceptional Families of Elements and the Solvability of Variational Inequalities for Unbounded Sets in Infinite Dimensional Hilbert Spaces”, Preprint, University of Beijing, 1999.Google Scholar
  48. Isac, G., and Zhao, YB. 1. Exceptional families of elements and the solvability of variational inequalities for unbounded sets in infinite dimensional Hilbert spaces. J. Math. Anal. Appl. 2000; 246:544–556.MathSciNetMATHCrossRefGoogle Scholar
  49. Kalashnikov, VV. “Complementarity Problem and Generalized Oligopoly Models”, Habilitation Thesis, Central Economics and Mathematics Institute, Moscow, Russia, 1995 (in Russian).Google Scholar
  50. Kalashnikov, VV. “Fixed Point Existence Theorems Based upon Topological Degree Theory”, Working Paper, Moscow, Central Economics and Mathematics Institute (in Russian), 1995.Google Scholar
  51. Kalashnikov VV and Kalashnikova NI. Solving two-level variational inequality. J.Global Optim. 1996; 8:289–294.MathSciNetMATHCrossRefGoogle Scholar
  52. Kalashnikov VV and Khan AA. A regularization approach for variational inequalities with pseudo-monotone operators. Operations Research Proceedings 1999, Selected papers of the Symposium on Operations Research (SOR’99); 1999 September 1–3; Magdeburg; Germany (K. Inderfurth e.a., eds.). Berlin- Heidelberg: Springer-Verlag, 2000.Google Scholar
  53. Karamardian S. 1. Generalized complementarity problem. J. Optim.Theory Appl. 1971; 8:161–168.MathSciNetMATHCrossRefGoogle Scholar
  54. Karamardian S. 2. Complementarity problems over cones with monotone and pseudomonolfone maps. J. Optim. Theory Appl. 1976; 18:445–454.MathSciNetMATHCrossRefGoogle Scholar
  55. Karamardian S, Schaible S. Seven kinds of monotone maps. J. Optim. Theory Appl. 1990; 66:37–46.MathSciNetMATHCrossRefGoogle Scholar
  56. Kinderlehrer, D., and Stampacchia, G. An Introduction to Variational Inequalities and Their Applications. New York: Academic Press, 1980.MATHGoogle Scholar
  57. Michael E. Selected selection theorems. Amer, Math. Monthly 1956; 63: 233–238.MathSciNetMATHCrossRefGoogle Scholar
  58. Mosco, U. 1. Implicit Variational Problems and Quasi-Variational Inequalities, Lecture Notes in Math., No. 543, Springer-Verlag, 1976.Google Scholar
  59. Mosco, U. 2. ‘On some nonlinear quasi-variational inequalities and implicit complementarity problems in stochastic control theory’. In: Variational Inequalities and Complementarity Problems, Theoryand Applications, R.W. Cottle, F. Giannessi and J.L. Lions, eds. New York: John Wiley and Sons, 1980.Google Scholar
  60. Nashed, M.Z., and Liu, F. ‘On nonlinear ill-posed problems II: monotone operator equations and monotone variational inequalities’. In: Theory and Applications of Nonlinear Operators of Accretive and Monotone Type, A.A. Karsatos, ed. Lecture Notes in Pure and Applied Mathematics, Vol.178, New York: Marcel Dekker, Inc., 1996.Google Scholar
  61. Nikaido, H. Convex Structures and Mathematical Economics. New York: Acad. Press, 1968.Google Scholar
  62. Outrata J.V. 1. On necessary optimality conditions for Stackelberg problems. J. Optim. Theory Appl. 1993; 76:305–320.MathSciNetMATHCrossRefGoogle Scholar
  63. Outrata J.V. 2. On optimization problems with variational inequality constraints. SIAM Journal on Opti mization 1994; 4:340–357.MathSciNetMATHCrossRefGoogle Scholar
  64. Pang, J.-S. 1. ‘The implicit complementarity problem’.- In: Nonlinear Programming IV, O.L. Mangasarian, R.R. Meyer and S.M. Robinson, eds. New York-London: Academic Press, 1981.Google Scholar
  65. Pang, J.-S. 2. On the convergence of a basic iterative method for the implicit complementarity problem. J.Optim.Theory Appl. 1982; 37:149–162.MathSciNetMATHCrossRefGoogle Scholar
  66. Rockafellar, R.T. Convex Analysis. Princeton, NJ: Prinston University Press, 1970.MATHGoogle Scholar
  67. Schaible, S. ‘Generalized monotonicity — concepts and uses’. In: Variational Inequalities and Network Equilibrium Problems, F. Giannessi and A. Maugeri, eds. New York: Plenum Publishing Corporation, 1995.Google Scholar
  68. Schaible S, Yao JC. On the equivalence of nonlinear complementarity problems and least-elements problems. Math. Programming 1995; 70:191–200.MathSciNetMATHCrossRefGoogle Scholar
  69. Sherali HD, Soyster AL and Murphy FH. Stackelberg-Nash-Cournot equilibria: characterizations and computations. Oper. Res. 1983; 31: 171–186.MathSciNetCrossRefGoogle Scholar
  70. Silin DB. Some properties of upper semi continuous multi-valued mappings. Proc. of Math. Inst, of the Russian Academy of Science (Steklov Institute) 1998; 185: 222–235 (in Russian). MathSciNetGoogle Scholar
  71. Smith MJ. A descent algorithm for solving monotone variational inequalities and monotone complementarity problems. J. Optim. Theory Appl. 1984; 44:485–496.MathSciNetMATHCrossRefGoogle Scholar
  72. Smith TE. A solution condition for complementarity problem with an application to spatial price equilibrium. Appl. Math, Comp. 1984; 15:61–69.MATHCrossRefGoogle Scholar
  73. Tikhonov, A.N., and Arsenin, V.Y. Solution of III-Posed Problems. New York: J.Wiley & Sons, 1977.Google Scholar
  74. Ton BA. Nonlinear operators on convex subsets of Banach spaces. Math. Ann. 1969; 181:35–44.MathSciNetMATHCrossRefGoogle Scholar
  75. Yao JC. Multivalued variational inequalities with X-pseudomonotone operators. J. Optim.Theory Appl. 1994; 83:391–403.MathSciNetMATHCrossRefGoogle Scholar
  76. Zeidler, E. Nonlinear Functional Analysis and its Applications, II/B. New York: Springer-Verlag, 1993.Google Scholar
  77. Zhao YB. 1. Exceptional family and finite-dimensional variational inequality over polyhedral convex set. Appl. Math. Comp. 1997; 87:111–126.MATHCrossRefGoogle Scholar
  78. Zhao YB. 2. “Existence Theory and Applications for Finite-Dimensional Variational Inequality and Complementarity Problems”, Ph.D. Thesis, Institute of Applied Mathematics, Academia Sinica, Beijing, China, 1998.Google Scholar
  79. Zhao YB, Han JY. Exceptional family of elements for a variational inequality problem and its applications. J. Global Opt. 1999; 14: 313–330.MathSciNetMATHCrossRefGoogle Scholar
  80. Zhao YB, Han JY, Qi HD. Exceptional family and existence theorems for variational inequality problems. J. Optim. Theory Appl. 1999;101: 475–495.MathSciNetMATHCrossRefGoogle Scholar
  81. Zhao, Y.B., and Isac, G. 1. “Properties of a Multivalued Mapping and Existence of Central Path for Some Nonmono-tone Complementarity Problems”, Preprint, University of Beijing, 1999.Google Scholar
  82. Zhao, Y.B., and Isac, G. 2. Quasi-P* and P(τ, α, β)-maps, exceptional family of elements and complementarity problems. J. Optim. Theory Appl. 2000; 105:213–231.MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2002

Authors and Affiliations

  • G. Isac
    • 1
  • V. A. Bulavsky
    • 2
  • V. V. Kalashnikov
    • 2
  1. 1.Department of Mathematics and Computer ScienceRoyal Military College of CanadaKingstonCanada
  2. 2.Central Economics Institute (CEMI) of Russian Academy of SciencesMoscowRussia

Personalised recommendations