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On stability analysis in mathematical programming

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Sensitivity, Stability and Parametric Analysis

Part of the book series: Mathematical Programming Studies ((MATHPROGRAMM,volume 21))

Abstract

Sufficient conditions for upper semicontinuity of approximate solutions and continuity of the values of mathematical programming problems with respect to data perturbations are obtained by using the variational convergence, thereby generalizing many known results. Upper and approximate lower semicontinuity of solutions and multipliers in infinite dimensional convex programming are obtained under gamma convergence of the data.

Work partially supported by I.M.A. del C.N.R., Genova.

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References

  1. J.P. Aubin, Mathematical methods of game and economic theory (North-Holland, Amsterdam, 1979).

    MATH  Google Scholar 

  2. V. Barbu and Th. Precupanu, Convexity and optimization in Banach spaces (Editura Academiei and Sijthof-Noordhoff, Bucaresti, 1978).

    Google Scholar 

  3. I. Behar, “Modifications in optimization and saddle point problem”, SIAM Journal on Control and Optimization 17 (1979) 673–688.

    Article  MATH  MathSciNet  Google Scholar 

  4. L. Boccardo, “Perturbazioni di funzionali del calcolo delle variazioni”, Ricerche di Matematica 29 (1980) 213–242.

    MATH  MathSciNet  Google Scholar 

  5. G.B. Dantzig, J. Folkman and N. Shapiro, “On the continuity of the minimum set of a continuous function”, Journal of Mathematical Analysis and Applications 17 (1967) 519–548.

    Article  MATH  MathSciNet  Google Scholar 

  6. E. De Giorgi, “Convergence problems for functionals and operators”, in: E. De Giorgi, E. Magenes, U. Mosco, eds., Proceedings of the international meeting on recent methods in nonlinear analysis, Rome, 1978 (Pitagora, Bologna, 1979) pp. 131–188.

    Google Scholar 

  7. E. De Giorgi and T. Franzoni, “Su un tipo di convergenza variazionale”, Rendiconti del Seminario Matematico di Brescia 3 (1979) 63–101.

    Google Scholar 

  8. J.P. Evans and F.J. Gould, “Stability in nonlinear programming”, Operations Research 18 (1970) 107–119.

    Article  MATH  MathSciNet  Google Scholar 

  9. A.V. Fiacco, “Convergence properties of local solutions of mathematical programming problems in general spaces”, Journal of Optimization Theory and Applications 13 (1974) 1–12.

    Article  MATH  MathSciNet  Google Scholar 

  10. A.V. Fiacco and W.P. Hutzler, “Basic results in the development of sensitivity and stability analysis in constrained mathematical programming”, The George Washington University, Technical Report T-407, 1979.

    Google Scholar 

  11. H.J. Greenberg and W.P. Pierskalla, “Extension of the Evans-Gould stability theorems for mathematical programs”, Operations Research 20 (1972) 143–153.

    Article  MATH  MathSciNet  Google Scholar 

  12. D.J. Hartfield and G.L. Curry, “Concerning the solution vector for parametric programming problems”, SIAM Journal of Applied Mathematics 26 (1974) 294–296.

    Article  Google Scholar 

  13. W.W. Hogan, “Point-to-set maps in mathematical programming”, SIAM Review 15 (1973) 591–603.

    Article  MATH  MathSciNet  Google Scholar 

  14. A.D. Joffe and V.M. Tihomirov, “Theory of extremal problems” (North-Holland, Amsterdam, 1979).

    Google Scholar 

  15. J.L. Kelley, General Topology (Van Nostrand, Princeton, 1955).

    MATH  Google Scholar 

  16. A. Kirsch, “Continuous perturbations of infinite optimization problems”, Journal of Optimization Theory and Applications 32 (1980) 171–182.

    Article  MATH  MathSciNet  Google Scholar 

  17. K. Kuratowski, Topology I (Academic Press, New York, 1966).

    Google Scholar 

  18. H.J. Langen, “Convergence of dynamic programming models”, preprint.

    Google Scholar 

  19. P. Marcellini, “Un teorema di passaggio al limite per la somma di funzioni convesse”, Bollettino dell’Unione Matematica Italiana 11 (1975) 107–124.

    MATH  MathSciNet  Google Scholar 

  20. M. Matzeu, “Su un tipo di continuità dell’operatore subdifferenziale”, Bollettino dell’Unione Matematica Italiana 14B (1971) 480–490.

    MathSciNet  Google Scholar 

  21. R. Meyer, “The validity of a family of optimization methods”, SIAM Journal on Control 8 (1970) 41–54.

    Article  MATH  Google Scholar 

  22. U. Mosco, “Convergence of convex sets and solutions of variational inequalities”, Advances in Mathematics 3 (1969) 510–585.

    Article  MATH  MathSciNet  Google Scholar 

  23. R.T. Rockafellar, “Level sets and continuity of conjugate convex functions”, Transactions of American Mathematical Society 123 (1966) 46–64.

    Article  MATH  MathSciNet  Google Scholar 

  24. R.T. Rockafellar, Convex analysis (Princeton University Press, Princeton, 1970).

    MATH  Google Scholar 

  25. T. Zolezzi, “On convergence of minima”, Bollettino dell’Unione Matematica Italiana 8 (1973) 246–257.

    MATH  MathSciNet  Google Scholar 

  26. T. Zolezzi, “Variational perturbations of linear optimal control problems”, in: Proceedings of the international meeting on recent methods in nonlinear analysis, Rome, 1978 (Pitagora, Bologna, 1979) pp. 131–188.

    Google Scholar 

  27. T. Zolezzi, Approximations and perturbations of minimum problems, in preparation.

    Google Scholar 

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Authors and Affiliations

  1. Istituto Matematico, Università di Genova, 16132, Genova, Italy

    T. Zolezzi

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  1. T. Zolezzi
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Anthony V. Fiacco

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© 1984 The Mathematical Programming Society, Inc.

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Zolezzi, T. (1984). On stability analysis in mathematical programming. In: Fiacco, A.V. (eds) Sensitivity, Stability and Parametric Analysis. Mathematical Programming Studies, vol 21. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0121220

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  • DOI: https://doi.org/10.1007/BFb0121220

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-00912-9

  • Online ISBN: 978-3-642-00913-6

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