A strong duality theorem for the minimum of a family of convex programs

  • J. M. Borwein
Contributed Papers


We produce a duality theorem for the minimum of an arbitrary family of convex programs. This duality theorem provides a single concave dual maximization and generalizes recent work in linear disjunctive programming. Homogeneous and symmetric formulations are studied in some detail, and a number of convex and nonconvex applications are given.

Key Words

Families of programs nonconvex duality disjunctive programming symmetric homogeneous duality 


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  1. 1.
    Balas, E.,A Note on Duality in Disjunctive Programming, Journal of Optimization Theory and Applications, Vol. 15, pp. 523–528, 1977.Google Scholar
  2. 2.
    Glassey, C. R.,Explicit Duality for Convex Homogeneous Programming, Mathematical Programming, Vol. 10, pp. 176–191, 1976.Google Scholar
  3. 3.
    Ekeland, I., andTemam, R. Convex Analysis and Variational Problems, North-Holland Publishing Company, Amsterdam, Holland, 1976.Google Scholar
  4. 4.
    Rockafellar, R. T.,Convex Analysis, Princeton University Press, Princeton, New Jersey, 1970.Google Scholar
  5. 5.
    Bertsekas, D. P.,Necessary and Sufficient Condition for a Penalty Method to be Exact, Mathematical Programming, Vol. 9, pp. 87–99, 1975.Google Scholar
  6. 6.
    Borwein, J. M.,The Minimum of a Family of Programs, Manheim Symposium Proceedings, Manheim, Germany, in Methods of Operations Research, Vol. 31, Section 1, pp. 99–111, 1979.Google Scholar
  7. 7.
    Schechter, M.,Sufficient Conditions for Duality in Homogeneous Programming, Journal of Optimization Theory and Applications, Vol. 23, pp. 389–400, 1977.Google Scholar
  8. 8.
    Dantzig, G. B., andWolfe, D.,The Decomposition Algorithm for Linear Programs, Studies in Optimization, Vol. 10, Mathematical Association of America, Washington, D.C., 1974.Google Scholar
  9. 9.
    Kall, P.,Stochastic Linear Programming, Springer-Verlag, New York, New York, 1976.Google Scholar
  10. 10.
    Balas, E.,Disjunctive Programming, Carnegie-Mellon University, Management Science Department, Report No. 415, 1978.Google Scholar
  11. 11.
    Kelley, J. E.,The Cutting Plane for Solving Convex Programs, SIAM Journal on Applied Mathematics, Vol. 8, pp. 703–712, 1960.Google Scholar
  12. 12.
    Owen, G.,Cutting Planes for Programs with Disjunctive Constraints, Journal of Optimization Theory and Applications, Vol. 11, pp. 49–55, 1973.Google Scholar
  13. 13.
    Jeroslow, R. G.,Cutting Planes for Complementarity Constraints, SIAM Journal on Control and Optimization, Vol. 16, pp. 56–62, 1978.Google Scholar
  14. 14.
    Greenberg, H. J., andPierskalla, W. P.,Stability Theorems for Infinitely Constrained Mathematical Programs, Journal of Optimization Theory and Applications, Vol. 16, pp. 409–427, 1975.Google Scholar

Copyright information

© Plenum Publishing Corporation 1980

Authors and Affiliations

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

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