Mathematical Programming

, Volume 116, Issue 1–2, pp 369–396 | Cite as

Subgradients of marginal functions in parametric mathematical programming

  • B. S. MordukhovichEmail author
  • N. M. Nam
  • N. D. Yen


In this paper we derive new results for computing and estimating the so-called Fréchet and limiting (basic and singular) subgradients of marginal functions in real Banach spaces and specify these results for important classes of problems in parametric optimization with smooth and nonsmooth data. Then we employ them to establish new calculus rules of generalized differentiation as well as efficient conditions for Lipschitzian stability and optimality in nonlinear and nondifferentiable programming and for mathematical programs with equilibrium constraints. We compare the results derived via our dual-space approach with some known estimates and optimality conditions obtained mostly via primal-space developments.


Variational analysis and optimization Nonsmooth functions and set-valued mappings Generalized differentiation Marginal and value functions Mathematical programming 

Mathematics Subject Classification (2000)

90C30 49J52 49J53 


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  1. 1.
    Aubin J.-P. (1984). Lipschitz behavior of solutions to convex minimization problems. Math. Oper. Res. 9: 87–111 zbMATHMathSciNetGoogle Scholar
  2. 2.
    Auslender A. (1979). Differential stability in nonconvex and nondifferentiable programming. Math. Progr. Study 10: 29–41 zbMATHMathSciNetGoogle Scholar
  3. 3.
    Auslender A., Teboulle M. (2003). Asymptotic Cones and Functions in Optimization and Variational Inequalities. Springer, New York zbMATHGoogle Scholar
  4. 4.
    Bonnans J.F., Shapiro A. (2000). Perturbation Analysis of Optimization Problems. Springer, New York zbMATHGoogle Scholar
  5. 5.
    Borwein J.M., Zhu Q.J. (2005). Techniques of Variational Analysis. Springer, New York zbMATHGoogle Scholar
  6. 6.
    Clarke F.H. (1983). Optimization and Nonsmooth Analysis. Wiley, New York zbMATHGoogle Scholar
  7. 7.
    Dien P.H., Yen N.D. (1991). On implicit function theorems for set-valued maps and their application to mathematical programming under inclusion constraints. Appl. Math. Optim. 24: 35–54 zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Gauvin J., Dubeau F. (1982). Differential properties of the marginal function in mathematical programming. Math. Progr. Study 19: 101–119 MathSciNetGoogle Scholar
  9. 9.
    Gauvin J., Dubeau F. (1984). Some examples and counterexamples for the stability analysis of nonlinear programming problems. Math. Progr. Study 21: 69–78 zbMATHMathSciNetGoogle Scholar
  10. 10.
    Gollan B. (1984). On the marginal function in nonlinear programming. Math. Oper. Res. 9: 208–221 zbMATHMathSciNetCrossRefGoogle Scholar
  11. 11.
    Ha T.X.D. (2005). Lagrange multipliers for set-valued problems associated with coderivatives. J. Math. Anal. Appl. 311: 647–663 zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Ioffe A.D., Tihomirov V.M. (1979). Theory of extremal problems. North-Holland Publishing Co., Amsterdam-New York zbMATHGoogle Scholar
  13. 13.
    Lucet, Y., Ye, J.J.: Sensitivity analysis of the value function for optimization problems with variational inequality constraints. SIAM J. Control Optim. 40, 699–723 (2001); Erratum. SIAM J. Control Optim. 41, 1315–1319 (2002)Google Scholar
  14. 14.
    Luo Z.Q., Pang J.-S., Ralph D. (1996). Mathematical Programs with Equilibrium Constraints. Cambridge University Press, Cambridge Google Scholar
  15. 15.
    Maurer H., Zowe J. (1979). First and second-order necessary and sufficient optimality conditions for infinite-dimensional programming problems. Math. Prog. 16: 98–110 zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Minchenko L.I. (2003). Multivalued analysis and differential properties of multivalued mappings and marginal functions. Optimization and related topics. J. Math. Sci. 116: 3266 zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Mordukhovich, B.S.: Sensitivity analysis in nonsmooth optimization. In: Field, D.A., Komkov, V. (eds.) Theoretical Aspects of Industrial Design, pp. 32–46, SIAM Publications (1992)Google Scholar
  18. 18.
    Mordukhovich B.S. (2006). Variational Analysis and Generalized Differentiation, I: Basic Theory. Springer, Berlin Google Scholar
  19. 19.
    Mordukhovich B.S. (2006). Variational Analysis and Generalized Differentiation, II: Applications. Springer, Berlin Google Scholar
  20. 20.
    Mordukhovich B.S., Nam N.M. (2005). Variational stability and marginal functions via generalized differentiation. Math. Oper. Res. 30: 800–816 zbMATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Mordukhovich B.S., Nam N.M., Yen N.D. (2007). Fréchet subdifferential calculus and optimality conditions in nondifferentiable programming. Optimization 55: 685–708 CrossRefMathSciNetGoogle Scholar
  22. 22.
    Mordukhovich B.S., Shao Y. (1996). Nonsmooth analysis in Asplund spaces. Trans. Am. Math. Soc. 348: 1230–1280 CrossRefMathSciNetGoogle Scholar
  23. 23.
    Outrata J.V., Koĉvara M., Zowe J. (1998). Nonsmooth Approach to Optimization Problems with Equilibrium Constraints. Kluwer, Dordrecht zbMATHGoogle Scholar
  24. 24.
    Phelps R.R. (1993). Convex Functions, Monotone Operators and Differentiability, 2nd edn. Springer, Berlin zbMATHGoogle Scholar
  25. 25.
    Robinson S.M. (1979). Generalized equations and their solutions, I: Basic theory. Math. Progr. Study 10: 128–141 zbMATHGoogle Scholar
  26. 26.
    Rockafellar R.T. (1982). Lagrange multipliers and subderivatives of optimal value functions in nonlinear programming. Math. Progr. Study 17: 28–66 zbMATHMathSciNetGoogle Scholar
  27. 27.
    Rockafellar R.T. (1985). Extensions of subgradient calculus with applications to optimization. Nonlinear Anal. 9: 665–698 zbMATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    Rockafellar R.T., Wets R.J.-B. (1998). Variational Analysis. Springer, Berlin zbMATHGoogle Scholar
  29. 29.
    Thibault L. (1991). On subdifferentials of optimal value functions. SIAM J. Control Optim. 29: 1019–1036 zbMATHCrossRefMathSciNetGoogle Scholar
  30. 30.
    Ye J.J. (2001). Multiplier rules under mixed assumptions of differentiability and Lipschitz continuity. SIAM J. Control Optim. 39: 1441–1460 zbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2007

Authors and Affiliations

  1. 1.Department of MathematicsWayne State UniversityDetroitUSA
  2. 2.Institute of MathematicsHanoiVietnam

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