Affine minorants minimizing the sum of convex functions

  • L. McLinden
Contributed Papers


It is often possible to replace a convex minimization problem by an equivalent one, in which each of the original convex functions is replaced by a suitably chosen affine minorant. In this paper we identify essentially the minimal conditions permitting this replacement, and also shed light on the close and complete link between such optimal affine minorants and certain optimal dual vectors. An application to the ordinary convex programming problem is included.

Key Words

Affine minorants Lagrange multipliers inf-convolution 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Rockafellar, R. T.,Convex Analysis, Princeton University press, Princeton, New Jersey, 1970.Google Scholar
  2. 2.
    Blair, C. E.,Convex Optimization and Lagrange Multipliers, University of Illinois at Urbana-Champaign, College of Commerce and Business Administration, Working Paper No. 407, 1977.Google Scholar
  3. 3.
    Rockafellar, R. T.,Extensions of Fenchel's Duality Theorem for Convex Functions, Duke Mathematical Journal, Vol. 33, pp. 81–89, 1966.Google Scholar
  4. 4.
    Stoer, J., andWitzgall, C.,Convexity and Optimization in Finite Dimensions, I, Springer-Verlag, Berlin, Germany, 1970.Google Scholar
  5. 5.
    Rockafellar, R. T.,Conjugate Duality and Optimization, Regional Conference Series in Applied Mathematics, No. 16, Society for Industrial and Applied Mathematics, Philadelphia, Pennsylvania, 1974.Google Scholar
  6. 6.
    Moreau, J.-J.,Fonctionelles Convexes, Collège de France, Paris, France, Mimeographed Lecture Notes, 1966–67.Google Scholar
  7. 7.
    Bourbaki, N.,Espaces Vectoriels Topologiques, Hermann, Paris, France, 1953.Google Scholar

Copyright information

© Plenum Publishing Corporation 1978

Authors and Affiliations

  • L. McLinden
    • 1
  1. 1.Department of MathematicsUniversity of Illinois at Urbana-ChampaignUrbana

Personalised recommendations