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A Fenchel-Rockafellar type duality theorem for maximization

Published online by Cambridge University Press:  17 April 2009

Ivan Singer
Ivan Singer
Affiliation:
Institutul Naţional pentru Creaţie Stiinţifică şi Tehnică, Bucureşti, Romania; Institutul de Matematică, Bucureşti, Romania.
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Abstract

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We prove that sup(f-h)(E) = sup(h*-f*)(E*), where f is a proper lower semi-continuous convex functional on a real locally convex space E, h: E → = [-∞, +∞] is an arbitrary-functional and, f*, h* are their convex conjugates respectively. When h = δG, the indicator of a bounded subset G of E, this yields a formula for sup f(G).

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 20 , Issue 2 , April 1979 , pp. 193 - 198
Copyright
Copyright © Australian Mathematical Society 1979

References

[1]Holmes, Richard B., A course on optimization and best approximation (Lecture Notes in Mathematics, 257. Springer-Verlag, Berlin, Heidelberg, New York, 1972).CrossRefGoogle Scholar
[2]Rockafellar, R.T., “Extension of Fenchel's duality theorem for convex functions”, Duke Math. J. 33 (1966), 81–89.CrossRefGoogle Scholar
[3]Singer, Ivan, “Some new applications of the Fenchel-Rockafellar duality theorem: Lagrange multiplier theorems and hyperplane theorems for convex optimization and best approximation”, Nonlinear Anal. 3 (1979), 239–248.CrossRefGoogle Scholar
[4]Singer, Ivan, “Maximization of lower semi-continuous convex functionals on bounded subsets of locally convex spaces. II: Quasi-Lagrangian duality theorems”, submitted.Google Scholar