### 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 language | English |
---|---|

Pages (from-to) | 637-689 |

Number of pages | 53 |

Journal | Kybernetika |

Volume | 48 |

Issue number | 4 |

Publication status | Published - 2012 |

### Fingerprint

### 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

*Kybernetika*,

*48*(4), 637-689.

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

Research output: Contribution to journal › Article

*Kybernetika*, vol. 48, no. 4, pp. 637-689.

}

TY - JOUR

T1 - Generalized minimizers of convex integral functionals, bregman distance, pythagorean identities

AU - Csiszár, I.

AU - Matúš, František

PY - 2012

Y1 - 2012

N2 - 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.

AB - 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.

KW - Bregman projection

KW - Conic core

KW - Convex duality

KW - Generalized exponential family

KW - Generalized primal/dual solutions

KW - Inference principles

KW - Maximum entropy

KW - Minimizing sequence

KW - Moment constraint

KW - Normal integrand

UR - http://www.scopus.com/inward/record.url?scp=84866016929&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84866016929&partnerID=8YFLogxK

M3 - Article

VL - 48

SP - 637

EP - 689

JO - Kybernetika

JF - Kybernetika

SN - 0023-5954

IS - 4

ER -