Applied Mathematics and Optimization

, Volume 24, Issue 1, pp 35–54 | Cite as

On implicit function theorems for set-valued maps and their application to mathematical programming under inclusion constraints

  • Pham Huy Dien
  • Nguyen Dong Yen


In this paper we establish some implicit function theorems for a class of locally Lipschitz set-valued maps and then apply them to investigate some questions concerning the stability of optimization problems with inclusion constraints. In consequence we have an extension of some of the corresponding results of Robinson, Aubin, and others.


System Theory Mathematical Method Mathematical Programming Implicit Function Theorem Inclusion Constraint 
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  1. 1.
    Aubin, J.-P. (1980), Further properties of Lagrange multipliers in nonsmooth optimization, Appl. Math. Optim., 6, 79–90.Google Scholar
  2. 2.
    Aubin, J,-P. (1982), Comportement Lipschitzien des solutions de problemes de minimization convexes, C. R. Acad. Sci. Paris, 295, 235–238.Google Scholar
  3. 3.
    Aubin, J.-P., Ekeland, I. (1984), Applied Nonlinear Analysis, Wiley-Interscience, New York.Google Scholar
  4. 4.
    Aubin, J.-P., Frankowska, H. (1984), On inverse function theorems for set-valued maps, Working Paper No. 48-68, IIASA, Laxenburg, Austria.Google Scholar
  5. 5.
    Clarke, F. H. (1983), Optimization and Nonsmooth Analysis, Wiley-Interscience, New York.Google Scholar
  6. 6.
    Dien, P. H., (1983), Locally Lipschitzian set-valued maps and generalized extremal problems, Acta Math. Vietnam., 8, 109–122.Google Scholar
  7. 7.
    Dien, P. H. (1985), On the regularity condition for the extremal problem under locally Lipschitz inclusion constraints, Appl. Math. Optim., 13, 151–161.Google Scholar
  8. 8.
    Dien, P. H., Sach, P. H. (1989), Further properties of the regularity of inclusion systems, Nonlinear Anal. T.M.A., 13, 1251–1267.Google Scholar
  9. 9.
    Ekeland, I. (1974), On the variational principle, J. Math. Anal. Appl., 47, 324–353.Google Scholar
  10. 10.
    Methlouthi, H. (1977), Sur le calcul differentiel multivoque, C. R. Acad. Sci. Paris, 284, 141–144.Google Scholar
  11. 11.
    Pschenichnyi, B. N. (1986), Implicit function theorems for set-valued maps, Dokl. Akad. Nauk. USSR, 291, 1063–1067 (in Russian).Google Scholar
  12. 12.
    Robinson, S. M. (1976), Regularity and stability for convex multivalued functions, Math. Oper. Res., 1, 130–143.Google Scholar
  13. 13.
    Robinson, S. M. (1976), Stability theory for systems of inequalities, Part II: Differentiable nonlinear systems, SIAM J. Numer. Anal., 13, 497–513.Google Scholar
  14. 14.
    Rockafellar, R. T. (1979), Clarke's tangent cone and boundaries of closed sets inR n, Nonlinear Anal., 3, 145–154.Google Scholar
  15. 15.
    Rockafellar, R. T. (1982), Lagrange multipliers and subderivatives of optimal value functions in nonlinear programming, Math. Programming Stud., 17, 28–66.Google Scholar
  16. 16.
    Rudin, W. (1973), Functional Analysis, McGraw-Hill, New York.Google Scholar
  17. 17.
    Yen, N. D., Dien, P. H. (1990), On Differential Estimations for Marginal Functions in Mathematical Programming Problems with Inclusion Constraints, Lecture Notes in Control and Information Sciences, Vol. 143, Springer-Verlag, Berlin, pp. 244–251.Google Scholar
  18. 18.
    Zowe, J., Kurcyusz, S. (1979), Regularity and stability for the mathematical programming problem in Banach spaces, Appl. Math. Optim., 5, 49–62.Google Scholar

Copyright information

© Springer-Verlag New York Inc. 1991

Authors and Affiliations

  • Pham Huy Dien
    • 1
  • Nguyen Dong Yen
    • 1
  1. 1.Institute of MathematicsHanoiVietnam

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