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

I. Csiszár, František Matúš

Research output: Contribution to journalArticle

13 Citations (Scopus)

Abstract

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
JournalKybernetika
Volume48
Issue number4
Publication statusPublished - 2012

Fingerprint

Pythagorean identity
Bregman Distance
Integral Functionals
Integrand
Minimizer
Constraint Qualifications
Convex Duality
Dual Solutions
Strictly Convex
Dual Problem
Value Function
Differentiable
Irregular
Smoothness
Moment
Term

Keywords

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

Cite this

Generalized minimizers of convex integral functionals, bregman distance, pythagorean identities. / Csiszár, I.; Matúš, František.

In: Kybernetika, Vol. 48, No. 4, 2012, p. 637-689.

Research output: Contribution to journalArticle

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