Skip to main content
SpringerLink
Log in

Generalized Monotone Equilibrium 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

This chapter is devoted to equilibrium problems and variational inequalities under generalized monotonicity assumptions on cost functions. We present basic existence and uniqueness results of solutions both for scalar and for vector problems. Relationships between generalized monotonicity properties of cost functions of these problems are also considered. Moreover, we describe basic approaches to construct iterative solution methods, including their convergence properties.

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. and Appl. 58, 1977, 1–10.

    Article  MATH  MathSciNet  Google Scholar 

  2. Ansari, Q.H. Vector equilibrium problems and vector variational inequalities, In: Vector Variational Inequalities and Vector Equilibria. Mathematical Theories, F. Giannessi, ed., Kluwer Academic Publishers, Dordrecht-Boston-London, 2000, 1–15.

    Google Scholar 

  3. Ansari, Q.H., Konnov, I.V., and Yao, J.C. On generalized vector equilibrium problems, Nonl. Anal. 47, 2001, 543–554.

    MathSciNet  Google Scholar 

  4. Ansari, Q.H., Oettli, W., and Schlãger, D. A generalization of vectorial equilibria, Math. Meth. Oper. Res. 46, 1997, 147–152.

    Article  Google Scholar 

  5. Antipin, A.S. On convergence of proximal methods to fixed points of extremal mappings and estimates of their rate of convergence, Comp. Maths. Math. Phys. 35, 1995, 539–551.

    MATH  MathSciNet  Google Scholar 

  6. Arrow, K.J. and Hahn, F.H. General competitive analysis, Holden Day, San Francisco, 1972.

    Google Scholar 

  7. Aubin, J.-P. Optima and equilibria, Springer-Verlag, Berlin, 1998.

    Google Scholar 

  8. Aussel, D., Corvellec, J.-N., and Lassonde, M. Subdifferential characterization of quasiconvexity and convexity, J. Convex Analysis 1, 1994, 195–201.

    MathSciNet  Google Scholar 

  9. Avriel, M., Diewert, W.E., Schaible, S., and Zang, I. Generalized convexity, Plenum Press, New York, 1988.

    Google Scholar 

  10. Baiocchi, C. and Capelo, A. Variational and quasivariational inequalities. Applications to free boundary problems, John Wiley and Sons, New York, 1984.

    Google Scholar 

  11. Bakushinskii, A.B. and Goncharskii, A.V. Ill-posed problems: Theory and applications, Kluwer Academic Publishers, Dordrecht, 1994.

    Google Scholar 

  12. Belen’kii, V.Z. and Volkonskii, V.A., Editors, Iterative methods in game theory and programming, Nauka, Moscow, 1974. (in Russian)

    Google Scholar 

  13. Berge, C. Topological spaces, Oliver and Boyd, Edinburgh, 1963.

    Google Scholar 

  14. Bertsekas, D.P. Constrained optimization and Lagrange multiplier methods, Academic Press, New York, 1982.

    Google Scholar 

  15. Bianchi, M. Pseudo P-monotone operators and variational inequalities, Research Report No. 6, Istituto di Econometrica e Matematica per le Decisioni Economiche, Universita Cattolica del Sacro Cuore, Milan, 1993.

    Google Scholar 

  16. Bianchi, M., Hadjisavvas, N., and Schaible, S. Vector equilibrium problems with generalized monotone bifunctions, J. Optim. Theory Appl. 92, 1997, 527–542.

    Article  MathSciNet  Google Scholar 

  17. Bianchi, M. and Pini, R. A note on equilibrium problems for KKM-maps, Quaderno No. 37, Istituto di Econometrica e Matematica per le Applicazioni Economiche, Finanziare ed Attuari, Milan, 2000.

    Google Scholar 

  18. Bianchi, M. and Schaible, S. Generalized monotone bifunctions and equilibrium problems, J. Optim. Theory Appl. 90, 1996, 31–43.

    Article  MathSciNet  Google Scholar 

  19. Billups, S.C. and Ferris, M.C. QPCOMP: A quadratic programming based solver for mixed complementarity problems, Math. Progr. 76, 1997, 533–562.

    Article  MathSciNet  Google Scholar 

  20. Blackwell, D. An analogue of the minimax theorem for vector payoffs, Pacific J. Math. 6, 1956, 1–8.

    MATH  MathSciNet  Google Scholar 

  21. Blum, E. and Oettli, W. Variational principles for equilibrium problems, In: Parametric Optimization and Related Topics III, J. Guddat, H.Th. Jongen, B. Kummer, and F.Nozicka eds., Peter Lang Verlag, Frankfurt am Main, 1993, 79–88.

    Google Scholar 

  22. Blum, E. and Oettli, W. From optimization and variational inequalities to equilibrium problems, The Math. Student 63, 1994, 127–149.

    MathSciNet  Google Scholar 

  23. Borwein, J.M. and Lewis, A.S. Partially finite convex programming, Part I: Quasi relative interiors in duality theory, Math. Progr. 57, 1992, 15–48.

    MathSciNet  Google Scholar 

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

    Google Scholar 

  25. Browder, F.E. Multivalued monotone nonlinear mappings and duality mappings in Banach spaces, Trans. Amer. Math. Soc. 71, 1965, 780–785.

    MATH  MathSciNet  Google Scholar 

  26. Bruck, R. On weak convergence of an ergodic iteration for the solution of variational inequalities for monotone operators in Hilbert space, J. Math. Anal. Appl. 61, 1977, 159–164.

    Article  MATH  MathSciNet  Google Scholar 

  27. Cambini, R. and Komlósi, S. On the scalarization of pseudoconvexity and pseudomonotonicity concepts for vector-valued functions, in: Generalized Convexity, Generalized Monotonicity, J.-P. Crouzeix, J.-E. Martinez-Legaz, and M. Volle, eds., Kluwer Academic Publishers, Dordrecht-Boston-London, 1998, 277–290.

    Google Scholar 

  28. Cambini, R. and Komlósi, S. On polar generalized monotonicity in vector optimization, Optimization 47, 2000, 111–121.

    MathSciNet  Google Scholar 

  29. Censor, Y., Lusem, A.N., and Zenios, S. An interior point method with Bregman functions for the variational inequality problem with paramonotone operators, Math. Progr. 81, 1999, 373–400.

    Google Scholar 

  30. Chadli, O., Chbani, Z., and Riahi, H. Equilibrium problems with generalized monotone bifunctions and applications to variational inequalities, J. Optim. Theory Appl. 105, 2000, 299–323.

    MathSciNet  Google Scholar 

  31. Chen, G.Y. Existence of solution for a vector variational inequality: An extension of the Hartmann-Stampacchia theorem, J. Optim. Theory Appl. 74, 1992, 445–456.

    MATH  MathSciNet  Google Scholar 

  32. Chen, G.Y. and Cheng, G.M. Vector variational inequality and vector optimization, Lecture Notes in Economics and Mathematical Systems 258, Springer-Verlag, Heidelberg, 1987, 408–416.

    Google Scholar 

  33. Chen, G. Y. and Craven, B.D. A vector variational inequality and optimization over an efficient set, Zeit. Oper. Res. 3, 1990, 1–12.

    MathSciNet  Google Scholar 

  34. Chen, G. Y. and Craven, B.D. Existence and continuity for vector optimization, J. Optim. Theory Appl. 81, 1994, 459–468.

    MathSciNet  Google Scholar 

  35. Chen, G.Y. and Yang, X.Q., Vector complementarity problem and its equivalences with weak minimal element in ordered spaces, J. Math. Anal. Appl. 153, 1990, 136–158.

    Article  MathSciNet  Google Scholar 

  36. Clarke, F.H. Optimization and nonsmooth analysis, John Wiley and Sons, New York, 1983.

    Google Scholar 

  37. Crouzeix, J.-P. Pseudomonotone variational inequality problems: Existence of solutions, Math. Progr. 78, 1997, 305–314.

    Article  MATH  MathSciNet  Google Scholar 

  38. Crouzeix, J.-P. Characterization of generalized convexity and generalized monotonicity: A survey, in: Generalized Convexity, Generalized Monotonicity, J.-P. Crouzeix, J.-E. Martinez-Legaz, and M. Volle, eds., Kluwer Academic Publishers, Dordrecht-Boston-London, 1998, 237–256.

    Google Scholar 

  39. Crouzeix, J.-P. and Ferland, J. A. Criteria for differentiate generalized monotone maps, Math. Progr. 75, 1996, 399–406.

    Article  MathSciNet  Google Scholar 

  40. Crouzeix, J.-P. and Hassouni, A. Quasimonotonicity of separable operators and monotonicity indices, SIAM J. Optim. 4, 1994, 649–658.

    Article  MathSciNet  Google Scholar 

  41. Crouzeix, J.-P. and Hassouni, A. Generalized monotonicity of a separable product of operators: The multivalued case, Set-Valued Analysis 3, 1995, 351–373.

    Article  MathSciNet  Google Scholar 

  42. Crouzeix, J.-P., Hassouni, A., Lahlou, A., and Schaible, S. Positive definite matrices, generarized monotonicity and linear complementarity problems, SIAM J. Matrix Anal. Appl. 22, 2000, 66–85.

    Article  MathSciNet  Google Scholar 

  43. Crouzeix, J.-P., Marcotte, P., and Zhu, D.L. Conditions ensuring the applicability of cutting plane methods for solving variational inequalities, Math. Progr. 88, 2000, 521–539.

    MathSciNet  Google Scholar 

  44. Crouzeix, J.-P., and Schaible, S. Generalized monotone affine maps, SIAM J. Matrix Anal. Appl. 17, 1996, 992–997.

    Article  MathSciNet  Google Scholar 

  45. Daniilidis, A. and Hadjisavvas, N. Variational inequalities with quasimonotone multivalued operators, Working Paper, Department of Mathematics, University of the Aegean, Samos, Greece, March 1995.

    Google Scholar 

  46. Daniilidis, A. and Hadjisavvas, N. Existence theorems for vector variational inequalities, Bull. Austral. Math. Soc. 54, 1996, 473–481.

    MathSciNet  Google Scholar 

  47. Daniilidis, A. and Hadjisavvas, N. Characterization of nonsmooth semistrictly quasiconvex and strictly quasiconvex functions, J. Optim. Theory Appl. 102, 1999, 525–536.

    Article  MathSciNet  Google Scholar 

  48. Daniilidis, A. and Hadjisavvas, N. Coercivity conditions and variational inequalities, Math. Progr. 86, 1999, 433–438.

    Article  MathSciNet  Google Scholar 

  49. Daniilidis, A. and Hadjisavvas, N. On the subdifferentials of quasiconvex and pseudoconvex functions and cyclic monotonicity, J. Math. Anal. Appl. 237, 1999, 30–42.

    Article  MathSciNet  Google Scholar 

  50. Daniilidis, A. and Hadjisavvas, N. On generalized cyclically monotone operators and proper quasimonotonicity, Optimization 47, 2000, 123–135.

    MathSciNet  Google Scholar 

  51. Dem’yanov, V.F. and Rubinov, A.M. Approximate methods for solving extremum problems, Leningrad University Press, Leningrade, 1968 (in Russian; Engl. transl. in Elsevier Science B.V., Amsterdam, 1970).

    Google Scholar 

  52. Denault, M. and Goffin, J.-L. On a primal-dual analytic center cutting plane method for variational inequalities, Comp. Optim. Appl.. 12, 1999, 127–155.

    MathSciNet  Google Scholar 

  53. Ding, X.P. Browder-Hartmann-Stampacchia variational inequalities, J. Math. Anal. Appl. 173, 1993, 577–587.

    Article  MATH  MathSciNet  Google Scholar 

  54. Ding, X.P. and Tarafdar, E. Generalized nonlinear variational inequalities with non monotone set-valued mappings, Appl. Math. Lett. 7, 1994, 5–11.

    Article  MathSciNet  Google Scholar 

  55. Ding, X.P. and Tarafdar, E. Generalized variational-like inequalities with pseudomonotone set-valued mappings, Arch. Math. 74, 2000, 302–313.

    Article  MathSciNet  Google Scholar 

  56. Duvaut, G. and Lions, J.-L. Les inéquations en mechanique et physique, Dunod, Paris, 1972.

    Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  58. Fan Ky, A minimax inequality and applications, In: Inequalities. III, O. Shisha, ed., Academic Press, New York, 1972, 103–113.

    Google Scholar 

  59. Fan Ky, Some properties of convex sets related to fixed point theorems, Math. Ann. 266, 1984, 519–537.

    Article  MATH  MathSciNet  Google Scholar 

  60. Fang, S.C. and Petersen, E.L. Generalized variational inequalities, J. Optim. Theory Appl.. 38, 1982, 363–383.

    Article  MathSciNet  Google Scholar 

  61. Farquharson, R. Sur une généralisation de la notion d’équilibrium, Compt. Rend. Acad. Sci. Paris 240, 1955, 46–48.

    MATH  MathSciNet  Google Scholar 

  62. Ferris, M.C. and Pang, J.-S. Engineering and economic applications of complementarity problems, SIAM Review 39, 1997, 669–713.

    Article  MathSciNet  Google Scholar 

  63. Fichera, G. Problemi elastostatici con vincoli unilaterali; il problema di Signorini con ambigue al contorno, Acad. Naz. Lincei. Mem. 8, 1964, 91–140.

    MathSciNet  Google Scholar 

  64. Fukushima, M. Merit functions for variational inequality and complementarity problems, In: Nonlinear Optimization and Applications, G. Di Pillo and F. Giannessi. Plenum Press, New York, 1996, 155–170.

    Google Scholar 

  65. Giannessi, F. Theorems of alternative, quadratic programs and complementarity problems, in: Variational Inequalities and Complementarity Problems, R.W. Cottle, F. Giannessi, and J.-L. Lions, eds., John Wiley and Sons, Chichester, 1980, 151–186.

    Google Scholar 

  66. Giannessi, F. On Minty variational principle, in: New Trends in Mathematical Programming, F. Giannessi, S. Komlósi, and T. Rapcsák, eds., Kluwer Academic Publishers, Dordrecht, 1998, 93–99.

    Google Scholar 

  67. Giannessi, F., Editor, Vector variational inequalities and vector equilibria. Mathematical theories, Kluwer Academic Publishers, Dordrecht-Boston-London, 2000.

    Google Scholar 

  68. Giannessi, F. and Maugeri, A., Editors, Variational inequalities and network equilibrium problems, Plenum Press, New York, 1995.

    Google Scholar 

  69. Giannessi, F. and Pellegrini, L., Image space analysis for vector optimization and variational inequalities. Scalarization, In: Proceedings of Combinatorial and Global Optimization Conference, Chania, May 1998, P. Pardalos, ed., Kluwer Academic Publishers, Dordrecht-Boston-London, to appear.

    Google Scholar 

  70. Gill, P.E., Murray, W., and Wright, M.H. Practical optimization, Academic Press, New York, 1981.

    Google Scholar 

  71. Glowinski, R., Lions, J.-L., and Trémolières, R. Analyse numerique des inéquations variationnelles, Dunod, Paris, 1976.

    Google Scholar 

  72. Goffin, J.-L., Marcotte, P., and Zhu, D.L. An analytic center cutting plane method for pseudomonotone variational inequalities, Oper. Res. Lett. 20, 1999, 1–6.

    MathSciNet  Google Scholar 

  73. Goh, C.J. and Yang, X.Q. Vector equilibrium problems and vector optimization, Eur. J. Oper. Res. 116, 1999, 615–628.

    Article  Google Scholar 

  74. Gol’shtein, E.G., Nemirovskii, A.S., and Nesterov, Yu.E. Level method, its extensions and applications, Ekon. Mat. Met. 31, 1995, 164–181. (in Russian)

    Google Scholar 

  75. Gol’shtein, E.G. and Tret’yakov, N.V. Modified Lagrangians and monotone maps in optimization, John Wiley and Sons, New York, 1996.

    Google Scholar 

  76. Gowda, M.S. Pseudomonotone and copositive star matrices, Linear Algebra Appl. 113, 1989, 107–118.

    MATH  MathSciNet  Google Scholar 

  77. Gowda, M.S. Affine pseudomonotone mappings and the linear complementarity problem, SIAM J. Matr. Anal. Appl. 11, 1990, 373–380.

    MATH  MathSciNet  Google Scholar 

  78. Granas, A. Sur quelques méthodes topologiques en analyse convexe, in: Méthodes Topologiques en Analyse Convexe, A. Granas, ed., Les Presses de l’Université de Montreal, Montreal, 1990, 151–186.

    Google Scholar 

  79. Hadjisavvas, N., Kravvaritis, D., Pantelidis, G., and Polyrakis, I. Nonlinear monotone operators with values in L(X, Y), J. Math. Anal. Appl. 140, 1989, 83–94.

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  82. Hadjisavvas, N. and Schaible, S. From scalar to vector equilibrium problems in the quasimonotone case, J. Optim. Theory Appl. 96, 1998, 297–309.

    Article  MathSciNet  Google Scholar 

  83. Hadjisavvas, N. and Schaible, S. Quasimonotonicity and pseudomonotonicity in variational inequalities and equilibrium problems, in: Generalized Convexity, Generalized Monotonicity, J.-P. Crouzeix, J.-E. Martinez-Legaz, and M. Volle, eds., Kluwer Academic Publishers, Dordrecht-Boston-London, 1998, 257–275.

    Google Scholar 

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

    MathSciNet  Google Scholar 

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

    Google Scholar 

  86. Hassouni, A. Quasimonotone multifunctions; applications to optimality conditions in quasiconvex programming, Num. Funct. Anal, and Optim. 13, 1992, 267–275.

    MATH  MathSciNet  Google Scholar 

  87. He, B. A class of projection and contraction methods for monotone variational inequalities, Appl. Math, and Optim. 35, 1997, 69–76.

    MATH  MathSciNet  Google Scholar 

  88. Jeyakumar, V., Oettli, W., and Natividad, M. A solvability theorem for a class of quasiconvex mappings with applications to optimization, J. Math. Anal. Appl. 179, 1993, 537–546.

    Article  MathSciNet  Google Scholar 

  89. John, R. Variational inequalities and pseudomonotone functions: some characterizations, in: Generalized Convexity, Generalized Monotonicity, J.-P. Crouzeix, J.-E. Martinez-Legaz, and M. Volle, eds., Kluwer Academic Publishers, Dordrecht-Boston-London, 1998, 291–301.

    Google Scholar 

  90. John, R. A note on Minty variational inequalities and generalized monotonicity, in: Generalized Convexity and Generalized Monotonicity, N. Hadjisavvas, J.-E. Martinez-Legaz, and M. Penot, eds., Lecture Notes in Economics and Mathematical Systems 502, Springer-Verlag, Berlin-Heidelberg-New York, 2001, 240–246.

    Google Scholar 

  91. Kachurovskii, R.I. On monotone operators and convex functionals, Usp. Matem. Nauk 15, 1960, 213–215.

    Google Scholar 

  92. Kantorovich, L.V. and Akilov, G.P. Functional analysis in normed spaces, Fizmatgiz, Moscow, 1959. (in Russian)

    Google Scholar 

  93. Kanzow, C. and Qi, H.-D. A QP-free constrained Newton-type method for variational inequality problems, Math. Progr. 85, 1999, 81–106.

    Article  MathSciNet  Google Scholar 

  94. Karamardian, S. Generalized complementarity problems, J. Optim. Theory Appl. 8, 1971, 161–167.

    Article  MATH  MathSciNet  Google Scholar 

  95. Karamardian, S. An existence theorem for the complementarity problem, J. Optim. Theory Appl. 19, 1976, 227–232.

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  98. Karamardian, S., Schaible, S., and Crouzeix, J.-P. Characterizations of generalized monotone maps, J. Optim. Theory Appl. 76, 1993, 399–413.

    Article  MathSciNet  Google Scholar 

  99. Kazmi, K.R. Existence of solutions for vector optimization, Appl. Math. Letters 9, 1996, 19–22.

    MATH  MathSciNet  Google Scholar 

  100. Kazmi, K.R. Some remarks on vector optimization problems, J. Optim. Theory Appl. 96, 1998, 133–138.

    Article  MATH  MathSciNet  Google Scholar 

  101. Kazmi, K.R. Existence of solutions for vector saddle points problems, In: Vector Variational Inequalities and Vector Equilibria. Mathematical Theories, F. Giannessi, ed., Kluwer Academic Publishers, Dordrecht-Boston-London, 2000, 267–275.

    Google Scholar 

  102. Kazmi, K.R. On vector equilibrium problem, Proc. Indian Acad. Sci. (Math. Sci.) 110, 2000, 213–223.

    MATH  MathSciNet  Google Scholar 

  103. Khobotov, E.N. Modification of the extragradient method for solving variational inequalities and certain optimization problems, USSR Comp. Maths. Math. Phys. 27, 1987, 120–127.

    MATH  MathSciNet  Google Scholar 

  104. Kinderlehrer, D. and Stampacchia, G. An introduction to variational inequalities and their applications, Academic Press, New York, 1980.

    Google Scholar 

  105. Kiwiel, K.C. Proximal level bandle methods for convex nondifferentiale optimization, saddle point problems and variational inequalities, Math. Progr. 69, 1995, 89–109.

    MATH  MathSciNet  Google Scholar 

  106. Knaster, B., Kuratowski, C., and Mazurkiewicz, S., Ein Beweis des Fixpunktsatzes für N Dimensionale Simplexe, Fund. Math. 14, 1929, 132–137.

    Google Scholar 

  107. Kneser, H. Sur le théorèms fondamental de la théorie des jeux, C. R. Acad. Sci. Paris 234, 1952, 2418–2420.

    MathSciNet  Google Scholar 

  108. Komlósi, S. Generalized monotonicity in nonsmooth analysis, In: Generalized Convexity, S. Komlósi, T. Rapcsák, and S. Schaible, eds., Springer-Verlag, Heidelberg, 1994, 263–275.

    Google Scholar 

  109. Komlósi, S. On the Stampacchia and Minty variational inequalities, in: Generalized Convexity and Optimization in Economic and Financial Decisions, G. Giorgi and F. Rossi, eds., Pitagora Editrice, Bologna, 1999, 231–260.

    Google Scholar 

  110. Konnov, I.V. Combined relaxation methods for finding equilibrium points and solving related problems, Russ. Math. (Iz. VUZ) 37, 1993, no.2, 44–51.

    MATH  MathSciNet  Google Scholar 

  111. Konnov, I.V. A combined method for variational inequalities, News-Lett. Math. Progr. Assoc. 4, 1993, 64. (in Russian)

    Google Scholar 

  112. Konnov, I.V. On combined relaxation method’s convergence rates, Russ. Math. (Iz. VUZ) 37, 1993, no. 12, 89–92.

    MATH  MathSciNet  Google Scholar 

  113. Konnov, I.V. Combined relaxation methods for solving vector equilibrium problems, Russ. Math. (Iz. VUZ) 39, 1995, no. 12, 51–59.

    MATH  MathSciNet  Google Scholar 

  114. Konnov, I.V. A general approach to finding stationary points and the solution of related problems, Comp. Maths. Math. Phys. 36, 1996, 585–593.

    MATH  MathSciNet  Google Scholar 

  115. Konnov, I.V. A class of combined iterative methods for solving variational inequalities, J. Optim. Theory Appl. 94, 1997, 677–693.

    Article  MATH  MathSciNet  Google Scholar 

  116. Konnov, I.V. On systems of variational inequalities, Russ. Math. (Iz. VUZ) 41, 1997, no. 12, 77–86.

    MATH  MathSciNet  Google Scholar 

  117. Konnov, I.V. Methods for solving finite-dimensional variational inequalities, DAS, Kazan, 1998. (in Russian)

    Google Scholar 

  118. Konnov, I.V. A combined relaxation method for variational inequalities with nonlinear constraints, Math. Progr. 80, 1998, 239–252.

    MATH  MathSciNet  Google Scholar 

  119. Konnov, I.V. Accelerating the convergence rate of a combined relaxational method, Comp. Maths. Math. Phys. 38, 1998, 49–56.

    MATH  MathSciNet  Google Scholar 

  120. Konnov, I.V. On the convergence of combined relaxation methods for variational inequalities, Optim. Meth. Software 80, 1998, 239–252.

    MATH  MathSciNet  Google Scholar 

  121. Konnov, I.V. On quasimonotone variational inequalities, J. Optim. Theory Appl. 99, 1998, 165–181.

    Article  MATH  MathSciNet  Google Scholar 

  122. Konnov, I.V. Combined relaxation methods for variational inequalities, Lecture Notes in Economics and Mathematical Systems 495, Springer-Verlag, Berlin-Heidelberg-New York, 2001.

    Google Scholar 

  123. Konnov, I.V. On vector equilibrium and vector variational inequality problems, in: Generalized Convexity and Generalized Monotonicity, N. Hadjisavvas, J.-E. Martinez-Legaz, and M. Penot, eds., Lecture Notes in Economics and Mathematical Systems 502, Springer-Verlag, Berlin-Heidelberg-New York, 2001, 247–263.

    Google Scholar 

  124. Konnov, I.V. Relatively monotone variational inequalities over product sets, Oper. Res. Letters 28, 2001, 21–26.

    Article  MATH  MathSciNet  Google Scholar 

  125. Konnov, I.V. and Schaible, S. Duality for equilibrium problems under generalized monotonicity, J. Optim. Theory Appl. 104, 2000, 395–408.

    Article  MathSciNet  Google Scholar 

  126. Konnov, I.V. and Yao, J.C. On the generalized vector variational inequality problem, J. Math. Anal. Appl. 206, 1997, 42–58.

    Article  MathSciNet  Google Scholar 

  127. Konnov, I.V. and Yao, J.C. Existence of solutions for generalized vector equilibrium problems, J. Math. Anal. Appl. 233, 1999, 328–335.

    Article  MathSciNet  Google Scholar 

  128. Korpelevich, G.M. Extragradient method for finding saddle points and other problems, Matecon 12, 1976, 747–756.

    MATH  Google Scholar 

  129. Lee, G.M., Kim, D.S., Lee, B.S., and Cho, S.J. Generalized vector variational inequalities and fuzzy extension, Appl. Math. Letters 6, 1993, 47–51.

    MathSciNet  Google Scholar 

  130. Lee, G.M., Kim, D.S., Lee, B.S., and Yen, N.D. Vector variational inequalities as a tool for studying vector optimization problems, Nonlin. Anal. 34, 1998, 745–765.

    Article  MathSciNet  Google Scholar 

  131. Lee, G.M. and Kum, S. Vector variational inequalities in a topological vector space, In: Vector Variational Inequalities and Vector Equilibria. Mathematical Theories, F. Giannessi, ed., Kluwer Academic Publishers, Dordrecht-Boston-London, 2000, 307–320.

    Google Scholar 

  132. Lemaréchal, C., Nemirovskii, A., Nesterov, Y. New variants of bundle methods Math. Progr. 69, 1995, 111–147.

    Google Scholar 

  133. Lemke, C.E. Bimatrix equilibrium points and mathematical programming, Manag. Sci. 11, 1965, 681–689.

    MATH  MathSciNet  Google Scholar 

  134. Levin, A.L. On an algorithm for the minimization of convex functions, Soviet Math. Doklady 6, 1965, 286–290.

    Google Scholar 

  135. Levin, V.L. Quasiconvex functions and quasimonotone operators, J. Convex Analysis 2, 1995, 167–172.

    MATH  Google Scholar 

  136. Lin, G.-H. and Xia, Z.-Q. Some convergence results for pseudomonotone variational inequalities, Pure Math. Appl. 10, 1999, 173–182.

    MathSciNet  Google Scholar 

  137. Lin, K.L., Yang, D.P., and Yao, J.C. Generalized vector variational inequalities, J. Optim. Theory Appl. 92, 1997, 117–125.

    Article  MathSciNet  Google Scholar 

  138. Lin, L.-J. and Park, S. On some generalized quasi-equilibrium problems, J. Math. Anal. Appl. 224, 1998, 167–181.

    Article  MathSciNet  Google Scholar 

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

    MathSciNet  Google Scholar 

  140. Luc, D.T. Theory of vector optimization, Lecture Notes in Mathematics 543, Springer-Verlag, Berlin, 1989.

    Google Scholar 

  141. Luc, D.T. Characterizations of quasiconvex functions, Bull. Austral. Math. Soc. 48, 1993, 393–406.

    MATH  MathSciNet  Google Scholar 

  142. Luc, D.T. Existence results for densely pseudomonotone variational inequalities, J. Math. Anal. Appl. 254, 2001, 291–308.

    Article  MATH  MathSciNet  Google Scholar 

  143. Luc, D.T. and Schaible, S. Generalized monotone nonsmooth maps, J. Convex Analysis 3, 1996, 195–205.

    MathSciNet  Google Scholar 

  144. Magnanti, T.L. and Perakis, G. A unifying geometric framework and complexity analysis for variational inequalities, Math. Progr. 71, 1995, 327–352.

    MathSciNet  Google Scholar 

  145. Marcotte, P. and Zhu, D.L. A cutting plane method for quasimonotone variational inequalities, Comp. Math. Appl. 20, 2001, 317–324.

    MathSciNet  Google Scholar 

  146. Martinet, B. Regularization d’inéquations variationnelles par approximations successives, Rev. Fr. Inf. Rech. Oper. 4, 1970, 154–159.

    MATH  MathSciNet  Google Scholar 

  147. Martos, B. Nonlinear programming. Theory and methods, Akadémiai Kiado, Budapest, 1975.

    Google Scholar 

  148. Minty, G. On the generalization of a direct method of the calculus of variations, Bull. Amer. Math. Soc. 73, 1967, 315–321.

    MATH  MathSciNet  Google Scholar 

  149. Mosco, U. Implicit variational problems and quasivariational inequalities, In: Nonlinear Operators and Calculus of Variations, J.P. Gossez et al., eds., Lecture Notes in Mathematics 543, Springer-Verlag, Berlin, 1976, 83–156.

    Google Scholar 

  150. Nagurney, A. Network economics: A variational inequality approach, Kluwer Academic Publishers, Dordrecht, 1993.

    Google Scholar 

  151. Nemirovskii, A.S. Effective iterative methods for solving equations with monotone operators, Ekon. Matem. Met. (Matecon) 17, 1981, 344–359.

    MATH  MathSciNet  Google Scholar 

  152. Nemirovskii, A.S. and Yudin, D.B. Problem complexity and method efficiency in optimization, Nauka, Moscow, 1979. (in Russian; Engl. transl. in John Wiley and Sons, New York, 1983).

    Google Scholar 

  153. von Neumann, J. and Morgenstern, O. J Game theory and economic behaviour, Princeton University Press, Princeton, 1953.

    Google Scholar 

  154. Nikaido, H. Convex structures and economic theory, Academic Press, New York, 1968.

    Google Scholar 

  155. Nikaido, H. and Isoda, K. Note on noncooperative convex games, Pacific J. Math. 5, 1955, 807–815.

    MathSciNet  Google Scholar 

  156. Nirenberg, L. Topics in nonlinear functional analysis, New York University Press, New York, 1974.

    Google Scholar 

  157. Oettli, W. A remark on vector-valued equilibria and generalized monotonicity, Acta Mathem. Vietnam. 22, 1997, 213–221.

    MATH  MathSciNet  Google Scholar 

  158. Oettli, W. and Schlãger, D. Generalized vectorial equilibria and generalized monotonicity, in Functional Analysis with Current Applications in Science, Technology and Industry, M. Brokate and A.H. Siddiqi, eds., Pitman Research Notes in Mathematics Series 377, Longman Publishing Co., Essex, 1998, 145–154.

    Google Scholar 

  159. Ortega, J.M. and Rheinboldt, W.C. Iterative solution of nonlinear equations in several variables, Academic Press, New York, 1970.

    Google Scholar 

  160. Patriksson, M. Nonlinear programming and variational inequality problems: A unified approach, Kluwer Academic Publishers, Dordrecht, 1999.

    Google Scholar 

  161. Pellegrini, L. Vector formulation of some economic equilibrium problems, in: Generalized Convexity and Optimization in Economic and Financial Decisions, G. Giorgi and F. Rossi, eds., Pitagora Editrice, Bologna, 1999, 323–331.

    Google Scholar 

  162. Penot, J.-P. and Quang, P.H. Generalized convexity of functions and generalized monotonicity of set-valued maps, J. Optim. Theory Appl. 92, 1997, 343–356.

    Article  MathSciNet  Google Scholar 

  163. Podinovskii, V.V. and Nogin, V.D. Pareto-optimal solutions of multicriteria problems, Nauka, Moscow, 1982 (in Russian).

    Google Scholar 

  164. Rockafellar, R.T. Convex analysis, Princeton University Press, Princeton, 1970.

    Google Scholar 

  165. Rockafellar, R.T. Monotone operators and the proximal point algorithm, SIAM J. Contr. Optim. 14, 1976, 877–898.

    MATH  MathSciNet  Google Scholar 

  166. Rockafellar, R.T. The theory of subgradients and its applications to problems of optimization, Heldermann-Verlag, Berlin, 1981.

    Google Scholar 

  167. Rosen, J.B. Existence and uniqueness of equilibrium points for concave n-person games, Econometrica 33, 1965, 520–534.

    MATH  MathSciNet  Google Scholar 

  168. Sach, P.H. Characterization of scalar quasiconvexity and convexity of locally Lipschitz vector valued maps, Optimization 46, 1999, 283–310.

    MATH  MathSciNet  Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

  170. Schaible, S. Generalized monotone maps, In: Proceedings of a Conference Held at “G. Stampacchia International School of Mathematics”, F. Giannessi, ed., Gordon and Breach Science Publishers, Amsterdam, 1992, 392–408.

    Google Scholar 

  171. Schaible, S. Generalized monotonicity-concepts and uses, in: Variational Inequalities and Network Equilibrium Problems, F. Giannessi and A. Maugeri, eds., Plenum Press, New York, 1995, 289–299.

    Google Scholar 

  172. Schaible, S. Criteria for generalized monotonicity, in: New Trends in Mathematical Programming, F. Giannessi, S. Komlósi, and T. Rapcsák, eds., Kluwer Academic Publishers, Dordrecht, 1998, 277–288.

    Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  174. Shor, N.Z. Cut-off method with space dilation in convex programming problems, Cybernetics 13, 1977, 94–96.

    Google Scholar 

  175. Shor, N.Z. New development trends in nonsmooth optimization methods, Cybernetics 6, 1977, 87–91.

    MATH  Google Scholar 

  176. Shor, N.Z. Minimization methods for non-differentiate functions, Naukova Dumka, Kiev, 1979. (in Russian; Engl. transl. in Springer-Verlag, Berlin, 1985).

    Google Scholar 

  177. Solodov, M.V. and Tseng, P. Modified projection-type methods for monotone variational inequalities, SIAM J. Contr. Optim. 34, 1996, 1814–1830.

    Article  MathSciNet  Google Scholar 

  178. Stampacchia, G. Formes bilinéaires coercitives sur le ensembles convexes, C. R. Acad. Sci. Paris 258, 1964, 4413–4416.

    MATH  MathSciNet  Google Scholar 

  179. Sun, D. A new step-size skill for solving a class of nonlinear projection equations, J. Comp. Math. 13, 1995, 357–368.

    MATH  Google Scholar 

  180. Sun, D. A class of iterative methods for solving nonlinear projection equations, J. Optim. Theory Appl. 91, 1996, 123–140.

    Article  MATH  MathSciNet  Google Scholar 

  181. Tanaka, T. Cone convexity of vector-valued functions, Sci. Rep. Hirosaki Univ. 37, 1990, 170–177.

    MATH  MathSciNet  Google Scholar 

  182. Tanaka, T. Generalized quasiconvexities, cone saddle points, and minimax theorems for vector-valued functions, J. Optim. Theory Appl. 81, 1994, 355–377.

    Article  MATH  MathSciNet  Google Scholar 

  183. Tanaka, T. Cone quasiconvexity of vector-valued functions, Sci. Rep. Hirosaki Univ. 42, 1995, 157–163.

    MATH  MathSciNet  Google Scholar 

  184. Tanino, T. and Sawaragi, Y. Duality theory in multiobjective optimization, J. Optim. Theory Appl. 27, 1979, 509–529.

    Article  MathSciNet  Google Scholar 

  185. Tikhonov, A.N. and Arsenin, A.V. Solution of ill-posed problems, John Wiley and Sons, New York, 1977.

    Google Scholar 

  186. Tseng, P. On linear convergence of iterative methods for the variational inequality problem, J. Comp. Appl. Math. 60, 1995, 237–252.

    Article  MATH  MathSciNet  Google Scholar 

  187. Uryas’yev, S.P. Adaptive algorithms of stochastic optimization and game theory, Nauka, Moscow, 1990. (in Russian)

    Google Scholar 

  188. Yang, X.Q. Vector variational inequality and its duality, Nonlin. Anal., Theory, Methods and Appl. 21, 1993, 869–877.

    MATH  Google Scholar 

  189. Yang, X.Q. Vector complementarity and minimal element problems, J. Optim. Theory Appl. 77, 1993, 483–495.

    Article  MATH  MathSciNet  Google Scholar 

  190. Yang, X.Q. and Goh, C.J. On vector variational inequalities: Applications to vector equilibria, J. Optim. Theory Appl. 95, 1997, 431–443.

    MathSciNet  Google Scholar 

  191. Yao, J.C. Variational inequalities with generalized monotone operators, Math. Oper. Res. 19, 1994, 691–705.

    MATH  MathSciNet  Google Scholar 

  192. Yao, J.C. Multi-valued variational inequalities with K-pseudomonotone operators, J. Optim. Theory Appl. 83, 1994, 391–403.

    Article  MATH  MathSciNet  Google Scholar 

  193. Yen, N.D. and Lee, G.M. On monotone and strongly monotone vector variational inequalities, in: Generalized Convexity and Optimization in Economic and Financial Decisions, G. Giorgi and F. Rossi, eds., Pitagora Editrice, Bologna, 1999, 347–358.

    Google Scholar 

  194. Yu, S.J. and Yao, J.C. On vector variational inequalities, J. Optim. Theory Appl. 89, 1996, 749–769.

    Article  MathSciNet  Google Scholar 

  195. Yuan, G.X.Z. The study of minimax inequalities and applications to economies and variational inequalities, Memoires of the American Mathematical Society 132, Number 625, American Mathematical Society, Providence, 1998.

    Google Scholar 

  196. Yudin, D.B. and Nemirovskii, A.S. Informational complexity and efficient methods for the solution of convex extremal problems, I, II, Matecon 13, 1977, 25–45; 550–559.

    Google Scholar 

  197. Zarantonello, E.H. Projections on convex sets in Hilbert space and spectral theory, in: Contributions to Nonlinear Functional Analysis, E.H. Zarantonello, ed., Academic Press, New York-London, 1971, 237–424.

    Google Scholar 

  198. Zhou, J.X., and Chen, G. On diagonal convexity conditions for problems in convex analysis and quasi-variational inequalities, J. Math. Anal. Appl. 132, 1988, 213–225.

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Applied Mathematics, Kazan State University, Kazan, 420008, RUSSIA

    Igor Konnov

  2. Informatics Problems Institute of AS RT, Kazan, 420012, RUSSIA

    Igor Konnov

Authors
  1. Igor Konnov
    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

Konnov, I. (2005). Generalized Monotone Equilibrium 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_13

Download citation

Publish with us

Policies and ethics

Search

Navigation