Generalized minimizers of convex integral functionals, bregman distance, pythagorean identities

Imre Csiszár, František Matúš

Research output: Contribution to journalArticle

13 Citations (Scopus)


Integral functionals based on convex normal integrands are minimized subject to finitely many moment constraints. The integrands are finite on the positive and infinite on the negative numbers, strictly convex but not necessarily differentiable. The minimization is viewed as a primal problem and studied together with a dual one in the framework of convex duality. The effective domain of the value function is described by a conic core, a modification of the earlier concept of convex core. Minimizers and generalized minimizers are explicitly constructed from solutions of modified dual problems, not assuming the primal constraint qualification. A generalized Pythagorean identity is presented using Bregman distance and a correction term for lack of essential smoothness in integrands. Results are applied to minimization of Bregman distances. Existence of a generalized dual solution is established whenever the dual value is finite, assuming the dual constraint qualification. Examples of 'irregular' situations are included, pointing to the limitations of generality of certain key results.

Original languageEnglish
Pages (from-to)637-689
Number of pages53
Issue number4
Publication statusPublished - Sep 14 2012



  • Bregman projection
  • Conic core
  • Convex duality
  • Generalized exponential family
  • Generalized primal/dual solutions
  • Inference principles
  • Maximum entropy
  • Minimizing sequence
  • Moment constraint
  • Normal integrand

ASJC Scopus subject areas

  • Software
  • Control and Systems Engineering
  • Theoretical Computer Science
  • Information Systems
  • Artificial Intelligence
  • Electrical and Electronic Engineering

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