Minimization of a convex functional on a class of sets in a measure space
type: Item
instance of: scholarly article
title: Minimization of a convex functional on a class of sets in a measure space
zbMATH Open document ID: 0706.90097
DOI: 10.1007/BF01157284
author: V. A. Kaminskij
published in: Mathematical Notes
publication date: 1990-01-01T00:00:00Z
review text: Let X be a Banach space with the sigma-field \(\Sigma\) of Borel sets and with an atomless finite measure \(\mu\). Let z(x) be a convex \(\mu\)- integrable functional and \(\Omega\) a class of sets \(\omega\) from \(\Sigma\) with property \(\mu (\omega)\geq \mu_ 0\) for some \(\mu_ 0<\mu (X)\). In this paper properties of the extremum problem: minimize the functional \[ \phi_ z(\omega)=\int_{\omega}z(x)\mu (d\omega)/\int_{\omega}\mu (d\omega) \] over \(\omega\in \Sigma\), is considered. Conditions for the existence of an optimal solution \(\omega^*\) are presented. For the special case of X a Hilbert space and z an affine functional, the problem is reduced to a quadratic programming problem in Hilbert space.
reviewed by: Riho Lepp
Mathematics Subject Classification ID: 90C48
Mathematics Subject Classification ID: 65K10
Mathematics Subject Classification ID: 90C25
Mathematics Subject Classification ID: 90C20
Mathematics Subject Classification ID: 90-08
zbMATH DE Number: 4160515
zbMATH Keywords: Banach space
zbMATH Keywords: atomless finite measure
MaRDI profile type: Publication
cites work: Optimal Search for and Mining of Underwater Mineral Resources
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cites work: Contiguity of Probability Measures
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full work available at URL: https://doi.org/10.1007/bf01157284
OpenAlex ID: W2038543371
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description: scientific article; zbMATH DE number 4160515