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On the lower semicontinuity of optimal sets in convex parametric optimization

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Part of the book series: Mathematical Programming Studies ((MATHPROGRAMM,volume 10))

Abstract

Regarding a special class of convex parametric problems sufficient conditions for the lower semicontinuity of the optimal solution sets are developed.

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References

  1. C. Berge, Topological spaces (Macmillan, New York, 1963).

    MATH  Google Scholar 

  2. 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 

  3. G. Debreu, Theory of value (John Wiley, New York, 1959).

    MATH  Google Scholar 

  4. J.P. Evans and F.J. Gould, “Stability in nonlinear programming”, Operations Research 18 (1970) 107–118.

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  6. J. Guddal, “Stability in convex quadratic parametric programming”, Mathematische Operations forschung und Statistik 7 (1976) 223–245.

    Google Scholar 

  7. J. Guddat and D. Klatte. “Qualitative stability in nonlinear optimization”, Proceedings of the IX. Internat. Symposium on Mathematical Programming, Budapest, 1976 (to appear).

    Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  9. P. Huard, “Optimization algorithms and point-to-set maps”, Mathematical Programming 8 (1975) 308–331.

    Article  MATH  MathSciNet  Google Scholar 

  10. D. Klatte, “Untersuchungen zur lokalen Stabilität konvexer parametrischer Optimierungsprobleme”, Dissertation (A), (Humboldt-Universität, Berlin, 1977).

    Google Scholar 

  11. B. Kummer, “Stability and weak duality in convex programming without regularity”, preprint (Humboldt-Universität, Berlin, 1978).

    Google Scholar 

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

    Article  MATH  Google Scholar 

  13. F. Nožička, J. Guddat, H. Hollatz und B. Bank, Theorie der linearen parametrischen Optimierung (Akademie-Verlag, Berlin, 1974).

    MATH  Google Scholar 

  14. S.M. Robinson and R.H. Day, “A sufficient condition for continuity of optimal sets in mathematical programming”, Journal, of Mathematical Analysis and Application 45 (1974) 506–511.

    Article  MATH  MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

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P. Huard

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© 1979 The Mathematical Programming Society

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Klatte, D. (1979). On the lower semicontinuity of optimal sets in convex parametric optimization. In: Huard, P. (eds) Point-to-Set Maps and Mathematical Programming. Mathematical Programming Studies, vol 10. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0120847

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  • DOI: https://doi.org/10.1007/BFb0120847

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-00797-2

  • Online ISBN: 978-3-642-00798-9

  • eBook Packages: Springer Book Archive

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