Results in Mathematics

, Volume 3, Issue 1–2, pp 235–248 | Cite as

Maximization of lower semi-continuous convex functional on bounded subsets of locally convex spaces. II: Quasi-Lagrangian duality theorems

  • Ivan Singer
Forschungsbeiträge Research Paper
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Abstract

We prove, for a proper lower semi-continuous convex functional ƒ on a locally convex space E and a bounded subset G of E, a formula for sup ƒ(G) which is symmetric to the Lagrange multiplier theorem for convex minimization, obtained in [7], with the difference that for sup ƒ(G) Lagrange multiplier functionals need not exist. When ƒ is also continuous we give some necessary conditions for g0 ∈ G to satisfy ƒ(g0) = sup ƒ(G). Also, we give some applications to deviations and farthest points. Finally, we show the connections with the “hyperplane theorems” of our previous paper [8].

Keywords

Lagrange Multiplier Convex Subset Penalty Term Duality Theorem Convex Space 
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References

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    C. Franchetti and I. Singer, Deviation and farthest points in normed linear spaces. Rev. roumaine math. pures et appl. 24 (1979), 373–381.MathSciNetMATHGoogle Scholar
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    I. Singer, Generalizations of methods of best approximation to convex optimization in locally convex spaces. II: Hyperplane theorems. J. Math. Anal. Appl. 69 (1979), 571–584 (to appear).MathSciNetMATHCrossRefGoogle Scholar
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    I. Singer, Some new applications of the Fenchel-Rockafellar duality theorem: Lagrange multiplier theorems and hyperplane theorems for convex optimization and best approximation. Nonlinear Anal., Theory, Methods Appl. 3 (1979), 239–248.MATHCrossRefGoogle Scholar
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    I. Singer, Maximization of lower semi-continuous convex functionals on bounded subsets of locally convex spaces. I: Hyperplane theorems. Appl. Math. Optimiz. (to appear).Google Scholar
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    I. Singer, A Fenchel-Rockafellar type duality theorem for maximization. Bull. Austral. Math. Soc. 20 (1979) (to appear).Google Scholar

Copyright information

© Birkhäuser Verlag, Basel 1980

Authors and Affiliations

  • Ivan Singer
    • 1
  1. 1.National Institute for Scientific and Technical Creation and Institute of MathematicsBucharestRomania

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