Hostname: page-component-586b7cd67f-t7czq Total loading time: 0 Render date: 2024-11-27T19:07:56.015Z Has data issue: false hasContentIssue false

A General Formula on the Conjugate of the Difference of Functions

Published online by Cambridge University Press:  20 November 2018

J.-B. Hiriart-Urruty*
Affiliation:
Laboratoire D'analyse Numérique Université Paul Sabatier 118, Route de Narbonne 31062 Toulouse Cedex, France
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Given an arbitrary function g :X → (-∞, +∞] and a lowersemicontinuous convex function h:X → (-∞, +∞], we give the general expression of the conjugate (g — h)* of g - h in terms of g* and h*. As a consequence, we get Toland's duality theorem:

Keywords

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 1986

References

References

1. Auchmuty, G., Duality for nonconvex variational principles, J. of Differential Equations, 50 (1983), pp. 80145.Google Scholar
2. Azencott, R., Grandes déviations et applications, Lecture Notes in Mathematics, 774 (1980), pp. 1176.Google Scholar
3. Ekeland, I. and Temam, R., Analyse convexe et problèmes variationnels, Bordas, Paris (1974).Google Scholar
4. Ellaia, R. and Hiriart-Urruty, J. B., The conjugate of the difference of convex functions, to appear.Google Scholar
5. Laurent, P. J., Approximation et Optimisation, Hermann, Paris (1972).Google Scholar
6. Moreau, J. J., Fonctionnelles convexes; Séminaire sur les équations aux dérivées partielles II, Collège de France (1966-1967).Google Scholar
7. Pshenichnyi, B. N., Leçons sur les jeux différentiels, Contrôle optimal et jeux différentiels, Cahier de TI.R.I.A. N°4 (1971).Google Scholar
8. Rockafellar, R. T., Conjugate duality and Optimization, Reg. Conf. Ser. in Appl. Math., vol. 16 SIAM Publications (1974).Google Scholar
9. Toland, J. F., Duality in nonconvex optimization, J. of Math. Analysis and Applications, 66 (1978), pp. 399415.Google Scholar
10. Toland, J. F., A duality principle for nonconvex optimization and the calculus of variations, Arch, for Rational Mechanics and Analysis, 71 (1979), pp. 4161.Google Scholar

Note Added on Proofs: The equalities (2.8) and (2.9) also appear in the following papers:

11. Singer, I., A Fenchel-Rockafeller type duality theorem for maximization, Bull, of the Austral. Math. Soc., 20(1979), pp. 193198.Google Scholar
12. Singer, I., Maximization of lower semi-continuous convex functionals on bounded subsets of locally convex spaces, II: Quasi-Lagrangian duality theorems, Result. Mathl., 3 (1980), pp. 235 — 248.Google Scholar