TY - JOUR

T1 - Extremum problems with total variation distance and their applications

AU - Charalambous, Charalambos D.

AU - Tzortzis, Ioannis

AU - Loyka, Sergey

AU - Charalambous, Themistoklis

PY - 2014

Y1 - 2014

N2 - The aim of this paper is to investigate extremum problems with pay-off being the total variation distance metric defined on the space of probability measures, subject to linear functional constraints on the space of probability measures, and vice-versa; that is, with the roles of total variation metric and linear functional interchanged. Utilizing concepts from signed measures, the extremum probability measures of such problems are obtained in closed form, by identifying the partition of the support set and the mass of these extremum measures on the partition. The results are derived for abstract spaces; specifically, complete separable metric spaces known as Polish spaces, while the high level ideas are also discussed for denumerable spaces endowed with the discrete topology. These extremum problems often arise in many areas, such as, approximating a family of probability distributions by a given probability distribution, maximizing or minimizing entropy subject to total variation distance metric constraints, quantifying uncertainty of probability distributions by total variation distance metric, stochastic minimax control, and in many problems of information, decision theory, and minimax theory.

AB - The aim of this paper is to investigate extremum problems with pay-off being the total variation distance metric defined on the space of probability measures, subject to linear functional constraints on the space of probability measures, and vice-versa; that is, with the roles of total variation metric and linear functional interchanged. Utilizing concepts from signed measures, the extremum probability measures of such problems are obtained in closed form, by identifying the partition of the support set and the mass of these extremum measures on the partition. The results are derived for abstract spaces; specifically, complete separable metric spaces known as Polish spaces, while the high level ideas are also discussed for denumerable spaces endowed with the discrete topology. These extremum problems often arise in many areas, such as, approximating a family of probability distributions by a given probability distribution, maximizing or minimizing entropy subject to total variation distance metric constraints, quantifying uncertainty of probability distributions by total variation distance metric, stochastic minimax control, and in many problems of information, decision theory, and minimax theory.

KW - Extremum probability measures

KW - signed measures

KW - total variation distance

UR - http://www.scopus.com/inward/record.url?scp=84906538588&partnerID=8YFLogxK

U2 - 10.1109/TAC.2014.2321951

DO - 10.1109/TAC.2014.2321951

M3 - Article

AN - SCOPUS:84906538588

SN - 0018-9286

VL - 59

SP - 2353

EP - 2368

JO - IEEE Transactions on Automatic Control

JF - IEEE Transactions on Automatic Control

IS - 9

M1 - 6810162

ER -