Advertisement

Existence of Solutions to Bilevel Variational Problems in Banach Spaces

  • Maria Beatrice Lignola
  • Jacqueline Morgan
Chapter
Part of the Nonconvex Optimization and Its Applications book series (NOIA, volume 58)

Keywords

Variational Inequality Equilibrium Problem Finite Dimensional Space Quasi Variational Inequality Quasivariational Inequality 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    C. Baiocchi, A. Capelo, Variational and quasivariational inequalities, applications to free boundary problems, John Wiley and Sons, New-York, 1984.Google Scholar
  2. [2]
    T. Basar, G.J. Olsder, Dynamic Noncooperative Games, Academic Press, New York, Second Edition, 1995.Google Scholar
  3. [3]
    E. Cavazzuti, J. Morgan, Well-posed saddle point problems, Optimization, Theory and Algorithms, J.B. Hiriart-Urruty, W. Oettli, J. Stoer Eds., Marcel Dekker, New-York (1978), 61–76.Google Scholar
  4. [4]
    X.P. Ding, K.K. Tan, Generalized variational inequalities and generalized quasi-variational inequalities, Journal of Mathematical Analysis and Applications 148 (1990), 497–508.CrossRefMathSciNetGoogle Scholar
  5. [5]
    A.L. Dontchev, T. Zolezzi, Well-Posed Optimization Problems, Lecture Notes in Mathematics 1543 (1993), Springer-Verlag, Berlin.Google Scholar
  6. [6]
    N. Hadjisavvas, S. Schaible, Quasimonotonicity and pseudo monotonicity in variational inequalities and equilibrium problems, Generalized convexity, generalized monotonicity:recent results (Luminy, 1996), Nonconvex Optim.Appl. 27 (1998), Kluwer Acad. Publ., Dordrecht, 257–275.Google Scholar
  7. [7]
    P.T. Harker, J.S. Pang, Finite-dimensional variational inequality and nonlinear complementarity problems: a survey of theory, algorithms and application, Mathematical Programming 48 (1990), 161–220.MathSciNetGoogle Scholar
  8. [8]
    P.T. Harker, S.C. Choi, Penalty function approach for mathematical programs with variational inequality constraints, Information and Decision Technologies 17 (1991), 41–50.MathSciNetGoogle Scholar
  9. [9]
    D. Kinderlehrer, G. Stampacchia, An introduction to variational inequality and their applications, Academic press, New-York, 1980.Google Scholar
  10. [10]
    G. Leitmann, On Generalized Stackelberg Strategies, Journal of Optimization, Theory and Applications 26 (1978), 637–643.CrossRefMathSciNetzbMATHGoogle Scholar
  11. [11]
    M.B. Lignola, J. Morgan, Well-Posedness for Optimization Problems with constraints defined by Variational Inequality having a unique solution, to appear on Journal of Global Optimization.Google Scholar
  12. [12]
    M.B. Lignola, J. Morgan, Existence of solutions to generalized bilevel programming problem, Multilevel Programming-Algorithms and Applications, Eds A. Migdalas, P.M. Pardalos and P. Varbrand, Kluwer Academic Publishers (1998), 315–332.Google Scholar
  13. [13]
    M.B. Lignola, J. Morgan, Stability for regularized bilevel programming problem, Journal of Optimization, Theory and Applications 93 (1997), n.3, 575–596.CrossRefMathSciNetGoogle Scholar
  14. [14]
    M.B. Lignola, J. Morgan, Convergence of Solutions of Quasivariational Inequalities and Applications, Topological Methods in Nonlinear Analysis 10 (1997), 375–385.MathSciNetGoogle Scholar
  15. [15]
    M.B. Lignola, J. Morgan, Topological existence and stability for Stackelberg problems, Journal of Optimization Theory and Applications 84 (1995), n.1, 145–169.CrossRefMathSciNetGoogle Scholar
  16. [16]
    M.B. Lignola. J. Morgan. Semicontinuity and Episemicontinuity: Equivalence and Applications, Bollettino dell’Unione Matematica Italiana 8B (1994), 1–16.MathSciNetGoogle Scholar
  17. [17]
    M.B. Lignola, J. Morgan, Approximate solutions to variational inequalities and Applications, Equilibrium Problems with Side Constraints Langrangean Theory and Duality, Eds F. Giannessi and A. Maugeri Le Matematiche 49 (1994), 281–293.Google Scholar
  18. [18]
    P. Loridan, J. Morgan, New results on approximate solutions in two level optimization, Optimization 20 (1989), 819–836.MathSciNetGoogle Scholar
  19. [19]
    Z.Q. Luo, J.S. Pang, D. Ralph, Mathematical Programs with Equilibrium Constraints, Cambridge University Press, 1996.Google Scholar
  20. [20]
    L. Mallozzi, J. Morgan, Hierarchical Systems with Weighted Reaction Set, Nonlinear Optimization and Applications, Ed. Di Pillo and F. Giannessi, Plenum Press New York and London (1996), 271–283.Google Scholar
  21. [21]
    P. Marcotte, D.L. Zhu, Exact and Inexact Penalty Methods for the Generalized Bilevel Programming Problem, Mathematical Programming 74 (1996), 141–157.CrossRefMathSciNetGoogle Scholar
  22. [22]
    J. Morgan, Constrained well-posed two-level optimization problem, Non-smooth optimization and related topics, F.H. Clarke, V.F. Demyanov, F. Giannessi, Eds., Plenum press, New-York (1989), 307–325.Google Scholar
  23. [23]
    M. Margiocco, F. Patrone, L. Pusillo, A new approach to Tikhonov well-posedness for Nash Equilibria, Optimization 40 (1997), 385–400.MathSciNetGoogle Scholar
  24. [24]
    U. Mosco, Implicit variational problems and quasi variational inequalities, Proc. Summer School (Bruxelles, 1975), Nonlinear Operators and the Calculus of Variations, Lecture Notes in Math. 543 (1976), Springer, Berlin, 83–156.Google Scholar
  25. [25]
    J.V. Outrata, On optimization problems with variational inequality constraint, Siam J. on Optimization 4 (1994), 334–357.MathSciNetGoogle Scholar
  26. [26]
    M.H. Shih, K.K. Tan, The Ky Fan Minimax Principle with Convex Sections and Variational Inequalities, Differential Geometry, Calculus of Variations, and Their Applications, edited by G.M. Rassias and T.M. Rassias, Marcel Dekker, New York (1985), 471–481.Google Scholar
  27. [27]
    H. von Stackelberg, Marktform und Gleichgewicht, Julius Springer, Vienna, 1934.Google Scholar
  28. [28]
    C.L. Yen, A minimax inequality and its applications to variational inequalities, Pacific J. Math. 97 (1981), 477–481.MathSciNetzbMATHGoogle Scholar
  29. [29]
    M.B. Lignola, J. Morgan, Convergences for Variational Inequalities and Generalized Variational Inequalities, Atti Seminario Matematico Fisico Universitá di Modena 45 (1997), 377–388.MathSciNetGoogle Scholar

Copyright information

© Kluwer Academic Publishers 2001

Authors and Affiliations

  • Maria Beatrice Lignola
    • 1
  • Jacqueline Morgan
    • 2
  1. 1.Dipartimento di Matematica e Applicazioni “R. Caccioppoli”Università degli Studi di Napoli “Federico II”Napoli
  2. 2.Dipartimento di Matematica e Applicazioni “R. Caccioppoli”Università degli Studi di Napoli “Federico II” Compl. universitario Monte S. AngeloNapoli

Personalised recommendations