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Norm duality for convex processes and applications

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Abstract

One can associate two norms with a Banach space convex process. These norms are dual to each other and the norm of a process agrees with the dual norm of its adjoint. This norm duality provides an extremely general and simple way of establishing surjectivity or boundedness properties of homogeneous (linear or convex) inequality systems.

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References

  1. Borwein, J. M.,Adjoint Process Duality, Mathematics of Operations Research, Vol. 8, pp. 403–434, 1983.

    Google Scholar 

  2. Isac, G.,Sur la Surjectivité des Processus Convexes, Preprint, 1980.

  3. Isac, G.,Processus Convexes et Inéquations dans des Espaces de Dimension Infinite, Preprint, 1982.

  4. Pomerol, J. C.,Optimization in Banach Space of Systems Involving Convex Processes, Lecture Notes in Control and Information Science, No. 38, Edited by R. F. Dzenick and F. Kozin, Springer-Verlag, New York, New York, 1982.

    Google Scholar 

  5. Robinson, S.,Normed Convex Processes, Transactions of the American Mathematical Society, Vol. 174, pp. 127–140, 1974.

    Google Scholar 

  6. Rockafellar, R. T.,Monotone Procesees of Convex and Concave Type, American Mathematical Society, Memoir No. 77, 1967.

  7. Borwein, J. M.,Convex Relations in Optimization and Analysis, Generalized Concavity in Optimization and Economics, Edited by S. Schaible and W. Ziemba, Academic Press, New York, New York, 1981.

    Google Scholar 

  8. Jameson, G. J.,Ordered Linear Spaces, Springer-Verlag, New York, New York, 1970.

    Google Scholar 

  9. Robinson, S.,Regularity and Stability for Convex Multivalued Functions, Mathematics of Operations Research, Vol. 1, pp. 130–143, 1976.

    Google Scholar 

  10. Urescu, C.,Multifunctions with Convex Closed Graph, Czechoslovak Mathematical Journal, Vol. 7, pp. 438–441, 1975.

    Google Scholar 

  11. Holmes, R. B.,Geometric Functional Analysis, Springer-Verlag, New York, New York, 1975.

    Google Scholar 

  12. Borwein, J. M.,Weak Tangent Cones and Optimization in Banach Spaces, SIAM Journal on Control and Optimization, Vol. 16, pp. 512–522, 1978.

    Google Scholar 

  13. Kurcyusz, S.,On Existence and Nonexistence of Lagrange Multipliers in Banach Space, Journal of Optimization Theory and Applications, Vol. 20, pp. 81–110, 1976.

    Google Scholar 

  14. Niewenhuis, J.,Another Application of Guignard's Generalized Kuhn-Tucker Conditions, Journal of Optimization Theory and Applications, Vol. 30, pp. 117–125, 1980.

    Google Scholar 

  15. Schirotzek, W.,On Farkas Type Theorems, Commentationes Mathematicae Universitatis Carolina, Vol. 22, pp. 1–14, 1981.

    Google Scholar 

  16. Borwein, J. M.,The Generalized Linear Complementarity Problem Treated without Fixed-Point Theory, Journal of Optimization Theory and Applications, Vol. 35, pp. 343–356, 1981.

    Google Scholar 

  17. Borwein, J. M.,A Lagrange Multiplier Theorem and a Sandwich Theorem for Convex Relations, Mathematica Scandinavica, Vol. 48, pp. 189–204, 1981.

    Google Scholar 

  18. Zowe, J.,The Open Mapping Theorem, American Mathematical Monthly, Vol. 89, pp. 458–460, 1982.

    Google Scholar 

  19. Altman, M.,Contractors and Contractor Direction Theory and Applications, Marcel Dekker, New York, New York, 1977.

    Google Scholar 

  20. Rockafellar, R. T.,Conjugate Duality and Optimization, SIAM Publications, Philadelphia, Pennsylvania, 1974.

    Google Scholar 

  21. Clarke, F. E.,Remarks on the Constraint Sets in Linear Programming, American Mathematical Monthly, Vol. 58, pp. 351–352, 1961.

    Google Scholar 

  22. Glover, B. M.,A Generalized Farkas Lemma with Applications to Quasidifferentiable Programming, Zeitschrift fur Operations Research, Vol. 26, pp. 125–141, 1982.

    Google Scholar 

  23. Zalinescu, C.,A Generalization of the Farkas Lemma and Application to Convex Programming, Journal of Mathematical Analysis and Applications, Vol. 66, pp. 651–678, 1978.

    Google Scholar 

  24. Pschenichnyi, B. N.,Necessary Conditions for an Extremum, Marcel Dekker, New York, New York, 1971.

    Google Scholar 

  25. Makarov, V. L., andRubinov, A. M.,Mathematical Theory of Economic Dynamics and Equilibrium, Springer-Verlag, New York, New York, 1979.

    Google Scholar 

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Authors and Affiliations

  1. Mathematics Department, Dalhousie University, Halifax, Nova Scotia

    J. M. Borwein (Professor)

Authors
  1. J. M. Borwein
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Additional information

Communicated by A. V. Fiacco

This research was partially supported by the Natural Sciences and Engineering Research Council of Canada.

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Borwein, J.M. Norm duality for convex processes and applications. J Optim Theory Appl 48, 53–64 (1986). https://doi.org/10.1007/BF00938589

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  • DOI: https://doi.org/10.1007/BF00938589

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