Norm duality for convex processes and applications

  • J. M. Borwein
Contributed Papers


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.

Key Words

Convex processes adjoint processes open-mapping theorem inequality systems optimality conditions surjectivity 


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  1. 1.
    Borwein, J. M.,Adjoint Process Duality, Mathematics of Operations Research, Vol. 8, pp. 403–434, 1983.Google Scholar
  2. 2.
    Isac, G.,Sur la Surjectivité des Processus Convexes, Preprint, 1980.Google Scholar
  3. 3.
    Isac, G.,Processus Convexes et Inéquations dans des Espaces de Dimension Infinite, Preprint, 1982.Google Scholar
  4. 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. 5.
    Robinson, S.,Normed Convex Processes, Transactions of the American Mathematical Society, Vol. 174, pp. 127–140, 1974.Google Scholar
  6. 6.
    Rockafellar, R. T.,Monotone Procesees of Convex and Concave Type, American Mathematical Society, Memoir No. 77, 1967.Google Scholar
  7. 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. 8.
    Jameson, G. J.,Ordered Linear Spaces, Springer-Verlag, New York, New York, 1970.Google Scholar
  9. 9.
    Robinson, S.,Regularity and Stability for Convex Multivalued Functions, Mathematics of Operations Research, Vol. 1, pp. 130–143, 1976.Google Scholar
  10. 10.
    Urescu, C.,Multifunctions with Convex Closed Graph, Czechoslovak Mathematical Journal, Vol. 7, pp. 438–441, 1975.Google Scholar
  11. 11.
    Holmes, R. B.,Geometric Functional Analysis, Springer-Verlag, New York, New York, 1975.Google Scholar
  12. 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. 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. 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. 15.
    Schirotzek, W.,On Farkas Type Theorems, Commentationes Mathematicae Universitatis Carolina, Vol. 22, pp. 1–14, 1981.Google Scholar
  16. 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. 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. 18.
    Zowe, J.,The Open Mapping Theorem, American Mathematical Monthly, Vol. 89, pp. 458–460, 1982.Google Scholar
  19. 19.
    Altman, M.,Contractors and Contractor Direction Theory and Applications, Marcel Dekker, New York, New York, 1977.Google Scholar
  20. 20.
    Rockafellar, R. T.,Conjugate Duality and Optimization, SIAM Publications, Philadelphia, Pennsylvania, 1974.Google Scholar
  21. 21.
    Clarke, F. E.,Remarks on the Constraint Sets in Linear Programming, American Mathematical Monthly, Vol. 58, pp. 351–352, 1961.Google Scholar
  22. 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. 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. 24.
    Pschenichnyi, B. N.,Necessary Conditions for an Extremum, Marcel Dekker, New York, New York, 1971.Google Scholar
  25. 25.
    Makarov, V. L., andRubinov, A. M.,Mathematical Theory of Economic Dynamics and Equilibrium, Springer-Verlag, New York, New York, 1979.Google Scholar

Copyright information

© Plenum Publishing Corporation 1986

Authors and Affiliations

  • J. M. Borwein
    • 1
  1. 1.Mathematics DepartmentDalhousie UniversityHalifax

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