A Note on Entropy Optimization

  • Pierre Maréchal


The natural workspaces arising in entropy optimization problems are considered. Simple conditions on the constraint functions result in their decomposability (which in turn makes it possible to conjugate the objective functional through the integral). When the objective functional is the standard entropy, it is shown that conjugacy through the integral holds (at least locally) without assuming decomposability of the workspace. This is applied to the computation of the value function of an entropy problem with constraints on the second order moments.


Measurable Function Maximum Entropy Order Moment Dual Solution Finite Measure 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Borwein, J. M. and Lewis, A. S. (1993) Partially-finite programming in L1 and the existence of maximum entropy estimates. SIAM J. Optim. 3(2), 248–267Google Scholar
  2. 2.
    Castaing, C. and Valadier, M. (1977) Convex Analysis and Measurable Multi-functions. Lecture Notes in Mathematics No 580. Springer-VerlagGoogle Scholar
  3. 3.
    Dacunha-Castelle, D. and Gamboa, F. (1990) Maximum d’entropie et problème des moments. Ann. Inst. H. Poincaré Probab. Statist. 26, 567–596Google Scholar
  4. 4.
    Decarreau, A., Hilhorst, D., Lemaréchal, C. and Navaza, J. (1992) Dual methods in entropy maximization. Application to some problems in crystallography. SIAM J. Optim. 2(2), 173–197CrossRefGoogle Scholar
  5. 5.
    Dieudonné, J. (1951) Sur les espaces de Köthe. J. Anal. Math. 1(1), 81–115CrossRefGoogle Scholar
  6. 6.
    Maréchal, P. and Lannes, A. (1997) Unification of some deterministic and probabilistic methods for the solution of linear inverse problems via the principle of maximum entropy on the mean. Inverse Problems 13, 135–151CrossRefGoogle Scholar
  7. 7.
    Maréchal, P. (to appear) On the Principle of Maximum Entropy on the Mean as a methodology for the regularization of inverse problems. Proceedings of the 7th Vilnius Conference on Probability Theory and Mathematical Statistics and 22nd European Meeting of StatisticiansGoogle Scholar
  8. 8.
    Maréchal, P. (1997) Sur la Régularisation des prote èmes mal-posés. Applications en science du signal et de l’image pour l’astrophysique. PhD. Thesis, Université Paul Sabatier — Toulouse, FranceGoogle Scholar
  9. 9.
    Rockafellar, R. T. (1974) Conjugate Duality and Optimization. SIAM Conference Board of the Mathematical Sciences Regional Conference Series in Applied Mathematics, No. 16CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • Pierre Maréchal
    • 1
  1. 1.Laboratoire ACSIOM, Département de MathématiquesUniversité de Montpellier 2Montpellier Cedex 5France

Personalised recommendations