Abstract

The definition of a Bregman function, given by Censor and Lent in 1981 on the basis of Bregman's seminal 1967 paper, was subsequently used in a plethora of research works as a tool for building sequential and inherently parallel feasibility and optimization algorithms. Solodov and Svaiter have recently shown that it is not "minimal". Some of its conditions can be derived from the others. In this note we illuminate this finding from a different perspective by presenting an alternative proof of the equivalence between the original and the simplified definitions of Bregman functions in which redundant conditions are eliminated. This implicitly shows that the seemingly different notion of Bregman functions recently introduced by Butnariu and Iusem, when transported to a proper setting in Rn\mathbb{R}^n, is equivalent to the original concept. The results established in this context are also used to resolve a problem in proximity function minimization encountered by Byrne and Censor.

Author information

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

Charles Byrne

Dept. of Mathematical Sciences, University of Massachusetts, Lowell, MA 01854, U.S.A.

byrnec@cs.uml.edu

Keywords

  • Bregman function
  • Bregman distance
  • Kullback-Leibler distance
  • modulus of total convexity
  • sequential consistency
  • totally convex function

References

  1. H. H. Bauschke, J. M. Borwein: Legendre functions and the method of random Bregman projections, Journal of Convex Analysis 4 (1997) 27–67.https://doi.org/10.68381/jca04002Google Scholar
  2. H. H. Bauschke, J. M. Borwein: Joint and separate convexity of the Bregman distance, in: Inherently Parallel Algorithms in Feasibility and Optimization and Their Applications, D. Butnariu, Y. Censor, S. Reich (eds.), Elesevier Science, Amsterdam, Netherlands (2001) 23–36.https://doi.org/10.1016/S1570-579X(01)80004-5Google Scholar
  3. L. M. Bregman: The relaxation method for finding the common point of convex sets and its application to the solution of problems in convex programming, USSR Computational Mathematics and Mathematical Physics 7 (1967) 200–217.https://doi.org/10.1016/0041-5553(67)90040-7Google Scholar
  4. D. Butnariu, Y. Censor, S. Reich: Iterative averaging of entropic projections for solving stochastic convex feasibility problems, Computational Optimization and Applications 8 (1997) 21–39.https://doi.org/10.1023/A:1008654413997Google Scholar
  5. D. Butnariu, A. N. Iusem: Totally Convex Function for Fixed Points Computation and Infinite Dimensional Optimization, Kluwer Academic Publishers, Dordrecht, Netherlands (2000).https://doi.org/10.1007/978-94-011-4066-9Google Scholar
  6. C. Byrne, Y. Censor: Proximity function minimization using multiple Bregman projections, with applications to split feasibility and Kullback-Leibler distance minimization, Annals of Operations Research 105 (2001) 77–98.https://doi.org/10.1023/A:1013349430987Google Scholar
  7. Y. Censor, A. Lent: An iterative row-action method for interval convex programming, Journal of Optimization Theory and Applications 34 (1981) 322–353.https://doi.org/10.1007/BF00934676Google Scholar
  8. Y. Censor, S. A. Zenios: Parallel Optimization: Theory, Algorithms and Applications, Oxford University Press, New York (1997).Google Scholar
  9. K. Kiwiel: Free-steering relaxation methods for problems with strictly convex costs and linear constraints, Mathematics of Operations Research 22 (1997) 326–349.https://doi.org/10.1287/moor.22.2.326Google Scholar
  10. R. T. Rockafellar: Convex Analysis, Princeton University Press, Princeton, New Jersey (1970).Google Scholar
  11. M. V. Solodov, B. F. Svaiter: An inexact hybrid generalized proximal point algorithm and some new results on the theory of Bregman functions, Mathematics of Operations Research 25 (2000) 214–230.https://doi.org/10.1287/moor.25.2.214.12222Google Scholar

Suggested citation

D. Butnariu, C. Byrne, Y. Censor. “Redundant Axioms in the Definition of Bregman Functions.” Journal of Convex Analysis 10 (2003), No. 1, 245–254. https://doi.org/10.68381/jca1013

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