Abstract

We consider a general class of problems of minimization of convex integral functionals (such as entropy maximization) subject to linear constraints. Under general assumptions, the minimizers are characterized. Our results improve previous literature on the subject in the following directions: – necessary and sufficient conditions for the shape of the minimizers are proved – without constraint qualification – under infinitely many linear constraints subject to natural integrability conditions (no topological restrictions). This paper extends previous results of the author by relaxing some integrability conditions on the constraint. As a consequence, the minimizers may admit a singular component. Our proofs mainly rely on convex duality.

Author information

Contact details are reproduced from the original publication and may be historical.

Christian Léonard

Equipe Modal-X. Université de Paris X, Bâtiment G. 200, Avenue de la République, 92001 Nanterre Cedex, France
and: Centre de Mathématiques Appliquées, École Polytechnique, 91128 Palaiseau, France

Christian.Leonard@u-paris10.fr

Keywords

  • Maximum entropy method
  • relative entropy
  • convex integral functionals
  • convex conjugacy

Mathematics Subject Classification

49K22, 52A41, 46B10

References

  1. J. M. Borwein, A. S. Lewis: Duality relationships for entropy-like minimization problems, SIAM J. Control and Optim. 29(2) (1991) 325–338.https://doi.org/10.1137/0329017Google Scholar
  2. J. M. Borwein, A. S. Lewis: On the convergence of moment problems, Trans. Amer. Math. Soc. 325(1) (1991) 249–271.https://doi.org/10.1090/S0002-9947-1991-1008695-8Google Scholar
  3. J. M. Borwein, A. S. Lewis: Convergence of best entropy estimates, SIAM J. Optim. 1(2) (1991) 191–205.https://doi.org/10.1137/0801014Google Scholar
  4. J. M. Borwein, A. S. Lewis: Partially-finite programming in L1 and the existence of maximum entropy estimates, SIAM J. Optim. 3(2) (1993) 248–267.https://doi.org/10.1137/0803012Google Scholar
  5. J. M. Borwein, A. S. Lewis: Strong rotundity and optimization, SIAM J. Optim. 4(1) (1994) 146–158.https://doi.org/10.1137/0804008Google Scholar
  6. P. Cattiaux, F. Gamboa: Large deviations and variational theorems for marginal problems, Bernoulli 5 (1999) 81–108.https://doi.org/10.2307/3318614Google Scholar
  7. I. Csiszár, F. Gamboa, E. Gassiat: MEM pixel correlated solutions for generalized moment and interpolation problems, IEEE Trans. Inform. Theory. 45(7) (1999) 2253–2270.https://doi.org/10.1109/18.796367Google Scholar
  8. I. Csiszár: I-divergence geometry of probability distributions and minimization problems, Ann. Probab. 3 (1975) 146–158.https://doi.org/10.1214/aop/1176996454Google Scholar
  9. I. Csiszár: Sanov property, generalized I-projection and a conditional limit theorem, Ann. Probab. 12 (1984) 768–793.https://doi.org/10.1214/aop/1176993227Google Scholar
  10. I. Csiszár: Generalized projections for non-negative functions, Acta Math. Hungar. 68(1-2) (1995) 161–185.https://doi.org/10.1007/BF01874442Google Scholar
  11. D. Dacunha-Castelle, F. Gamboa: Maximum d’entropie et problèmes de moments, Ann. Inst. Henri Poincaré 26 (1990) 567–596.Google Scholar
  12. A. Dembo, O. Zeitouni: Large Deviations Techniques and Applications, Springer-Verlag (1998).https://doi.org/10.1007/978-1-4612-5320-4Google Scholar
  13. A. Fougères, E. Giner: Applications de la décomposition d’un espace d’Orlicz ..., Séminaire d’Analyse Convexe, Montpellier 76, Exposé 18 (1976).Google Scholar
  14. F. Gamboa, E. Gassiat: Bayesian methods and maximum entropy for ill posed problems, Ann. Statist. 25(1) (1997) 328–350.https://doi.org/10.1214/aos/1034276632Google Scholar
  15. E. Giner: Décomposabilité et dualité des espaces d’Orlicz, Séminaire d’Analyse Convexe, Montpellier 76, Exposé 16 (1976).Google Scholar
  16. A. Kozek: Convex integral functionals on Orlicz spaces, Commentat. Math. 21 (1979) 109–135.Google Scholar
  17. C. Léonard: Convex conjugates of integral functionals, Acta Math. Hungar. 93(4) (2001) 253–280.https://doi.org/10.1023/A:1017943805248Google Scholar
  18. C. Léonard: Minimizers of energy functionals, Acta Math. Hungar. 93(4) (2001) 281–325.https://doi.org/10.1023/A:1017919422086Google Scholar
  19. C. Léonard: Minimization of energy functionals applied to some inverse problems, J. Appl. Math. Optim. 44 (2001) 273–297.https://doi.org/10.1007/s00245-001-0019-5Google Scholar
  20. C. Léonard: Large deviations for Poisson random measure and processes with independent increments, Stoch. Proc. and Appl. 85 (2000) 93–121.https://doi.org/10.1016/S0304-4149(99)00067-8Google Scholar
  21. C. Léonard: Some results on entropic projections, Proceedings of the Conference on Stochastic Analysis and Mathematical Physics, Lisbon 1999, A. B. Cruzeiro, J. C. Zambrini (eds.), Progress in Probability, Birkhäuser (2001).https://doi.org/10.1007/978-1-4612-0127-4_4Google Scholar
  22. C. Léonard, J. Najim: An extension of Sanov’s theorem. Application to the Gibbs conditioning principle, Bernoulli (2002), to appear.Google Scholar
  23. F. Liese, I. Vajda: Convex Statistical Distances, B. G. Teubner, Leipzig (1987).Google Scholar
  24. R. T. Rockafellar: Integrals which are convex functionals II, Pacific J. Math. 39(2) (1971) 439–469.https://doi.org/10.2140/pjm.1971.39.439Google Scholar
  25. R. T. Rockafellar: Convex Analysis, Princeton University Press (1970).https://doi.org/10.1515/9781400873173Google Scholar
  26. M. Teboulle, I. Vajda: Convergence of best φ-entropy estimates, IEEE Trans. Inform. Theory 39 (1993) 297–301.https://doi.org/10.1109/18.179378Google Scholar

Suggested citation

C. Léonard. “Minimizers of Energy Functional under not very Integrable Constraints.” Journal of Convex Analysis 10 (2003), No. 1, 63–88. https://doi.org/10.68381/jca1003

Download BibTeXPublished by Heldermann Verlag.

Published by Heldermann Verlag, 2003. Rights now held by Banach Press LLC.