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Subgradients of marginal functions in parametric mathematical programming

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Abstract

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.

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References

  1. Aubin J.-P. (1984). Lipschitz behavior of solutions to convex minimization problems. Math. Oper. Res. 9: 87–111

    MATH  MathSciNet  Google Scholar 

  2. Auslender A. (1979). Differential stability in nonconvex and nondifferentiable programming. Math. Progr. Study 10: 29–41

    MATH  MathSciNet  Google Scholar 

  3. Auslender A., Teboulle M. (2003). Asymptotic Cones and Functions in Optimization and Variational Inequalities. Springer, New York

    MATH  Google Scholar 

  4. Bonnans J.F., Shapiro A. (2000). Perturbation Analysis of Optimization Problems. Springer, New York

    MATH  Google Scholar 

  5. Borwein J.M., Zhu Q.J. (2005). Techniques of Variational Analysis. Springer, New York

    MATH  Google Scholar 

  6. Clarke F.H. (1983). Optimization and Nonsmooth Analysis. Wiley, New York

    MATH  Google Scholar 

  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

    Article  MATH  MathSciNet  Google Scholar 

  8. Gauvin J., Dubeau F. (1982). Differential properties of the marginal function in mathematical programming. Math. Progr. Study 19: 101–119

    MathSciNet  Google Scholar 

  9. Gauvin J., Dubeau F. (1984). Some examples and counterexamples for the stability analysis of nonlinear programming problems. Math. Progr. Study 21: 69–78

    MATH  MathSciNet  Google Scholar 

  10. Gollan B. (1984). On the marginal function in nonlinear programming. Math. Oper. Res. 9: 208–221

    Article  MATH  MathSciNet  Google Scholar 

  11. Ha T.X.D. (2005). Lagrange multipliers for set-valued problems associated with coderivatives. J. Math. Anal. Appl. 311: 647–663

    Article  MATH  MathSciNet  Google Scholar 

  12. Ioffe A.D., Tihomirov V.M. (1979). Theory of extremal problems. North-Holland Publishing Co., Amsterdam-New York

    MATH  Google Scholar 

  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. Luo Z.Q., Pang J.-S., Ralph D. (1996). Mathematical Programs with Equilibrium Constraints. Cambridge University Press, Cambridge

    Google Scholar 

  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

    Article  MATH  MathSciNet  Google Scholar 

  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

    Article  MATH  MathSciNet  Google Scholar 

  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)

  18. Mordukhovich B.S. (2006). Variational Analysis and Generalized Differentiation, I: Basic Theory. Springer, Berlin

    Google Scholar 

  19. Mordukhovich B.S. (2006). Variational Analysis and Generalized Differentiation, II: Applications. Springer, Berlin

    Google Scholar 

  20. Mordukhovich B.S., Nam N.M. (2005). Variational stability and marginal functions via generalized differentiation. Math. Oper. Res. 30: 800–816

    Article  MATH  MathSciNet  Google Scholar 

  21. Mordukhovich B.S., Nam N.M., Yen N.D. (2007). Fréchet subdifferential calculus and optimality conditions in nondifferentiable programming. Optimization 55: 685–708

    Article  MathSciNet  Google Scholar 

  22. Mordukhovich B.S., Shao Y. (1996). Nonsmooth analysis in Asplund spaces. Trans. Am. Math. Soc. 348: 1230–1280

    Article  MathSciNet  Google Scholar 

  23. Outrata J.V., Koĉvara M., Zowe J. (1998). Nonsmooth Approach to Optimization Problems with Equilibrium Constraints. Kluwer, Dordrecht

    MATH  Google Scholar 

  24. Phelps R.R. (1993). Convex Functions, Monotone Operators and Differentiability, 2nd edn. Springer, Berlin

    MATH  Google Scholar 

  25. Robinson S.M. (1979). Generalized equations and their solutions, I: Basic theory. Math. Progr. Study 10: 128–141

    MATH  Google Scholar 

  26. Rockafellar R.T. (1982). Lagrange multipliers and subderivatives of optimal value functions in nonlinear programming. Math. Progr. Study 17: 28–66

    MATH  MathSciNet  Google Scholar 

  27. Rockafellar R.T. (1985). Extensions of subgradient calculus with applications to optimization. Nonlinear Anal. 9: 665–698

    Article  MATH  MathSciNet  Google Scholar 

  28. Rockafellar R.T., Wets R.J.-B. (1998). Variational Analysis. Springer, Berlin

    MATH  Google Scholar 

  29. Thibault L. (1991). On subdifferentials of optimal value functions. SIAM J. Control Optim. 29: 1019–1036

    Article  MATH  MathSciNet  Google Scholar 

  30. Ye J.J. (2001). Multiplier rules under mixed assumptions of differentiability and Lipschitz continuity. SIAM J. Control Optim. 39: 1441–1460

    Article  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Wayne State University, Detroit, MI, 48202, USA

    B. S. Mordukhovich & N. M. Nam

  2. Institute of Mathematics, 18 Hoang Quoc Viet Rd., 10307, Hanoi, Vietnam

    N. D. Yen

Authors
  1. B. S. Mordukhovich
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  2. N. M. Nam
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  3. N. D. Yen
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Corresponding author

Correspondence to B. S. Mordukhovich.

Additional information

Dedicated to Alfred Auslender in honor of his 65th birthday

Research was partially supported by the National Science Foundation under grant DMS-0304989 and by the Australian Research Council under grant DP-0451168.

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Mordukhovich, B.S., Nam, N.M. & Yen, N.D. Subgradients of marginal functions in parametric mathematical programming. Math. Program. 116, 369–396 (2009). https://doi.org/10.1007/s10107-007-0120-x

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  • DOI: https://doi.org/10.1007/s10107-007-0120-x

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