Mathematical Programming

, Volume 13, Issue 1, pp 183–199 | Cite as

Multivalued convexity and optimization: A unified approach to inequality and equality constraints

  • J. Borwein


Multivalued functions satisfying a general convexity condition are examined in the first section. The second section establishes a general transposition theorem for such functions and develops an abstract multiplier principle for them. In particular both convex inequality and linear equality constraints are seen to satisfy the same generalized constraint qualification. The final section examines quasi-convex programmes.

Key words

Multivalued convexity Quasi-convexity Convex programming Alternative theorems Generalized Slater condition Transposition theorems Lagrange multipliers 


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  1. [1]
    R.A. Abrams and A. Ben-Israel, “Optimality conditions and recession cones”,Operations Research 23 (1975) 549–553.Google Scholar
  2. [2]
    C. Berge,Espace topologique, fonction multivoque (Dunod, Paris, 1959).Google Scholar
  3. [3]
    B.D. Craven and B. Mond, “Transposition theorems for cone-convex functions”,SIAM Journal of Applied Mathematics 24 (1973) 603–612.Google Scholar
  4. [4]
    J.L. Kelley and I. Namioka,Linear topological spaces (Van Nostrand, New York, 1963).Google Scholar
  5. [5]
    R. Lehmann and W. Oettli, “The theorem of the alternative, the key theorem, and the vector maximization problem”,Mathematical Programming 8 (1975) 332–344.Google Scholar
  6. [6]
    D.G. Luenberger,Optimization by vector space methods (J. Wiley, New York, 1969).Google Scholar
  7. [7]
    D.G. Luenberger, “Quasi-convex programming”,SIAM Journal of Applied Mathematics 16 (1968) 1090–1095.Google Scholar
  8. [8]
    O.L. Mangasarian,Nonlinear programming (McGraw-Hill, New York, 1969).Google Scholar
  9. [9]
    T.S. Motzkin, “Beiträge zur Theorie der linearen ungleichungen”, Dissertation, University of Basel (Jerusalem, 1936).Google Scholar
  10. [10]
    J. Ponstein, “Seven kinds of convexity”,SIAM Review 2 (1967) 115–119.Google Scholar
  11. [11]
    A.P. Robertson and W.J. Robertson,Topological vector spaces (Cambridge University Press, Cambridge 1964).Google Scholar
  12. [12]
    S.M. Robinson, “Regularity and stability for convex multivalued functions”,Mathematics of Operations Research 1 (1976) 130–143.Google Scholar
  13. [13]
    R.T. Rockafellar,Convex analysis (Princeton University Press, 1970).Google Scholar
  14. [14]
    J. Stoer and C. Witzgall,Convexity and optimization in finite dimensions, I (Springer, Berlin, 1970).Google Scholar

Copyright information

© The Mathematical Programming Society 1977

Authors and Affiliations

  • J. Borwein
    • 1
  1. 1.Dalhousie UniversityHalifaxCanada

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