Minimization of quadratic functionals with constraints in the form of equalities
type: Item
instance of: scholarly article
title: Minimization of quadratic functionals with constraints in the form of equalities
zbMATH Open document ID: 0961.49013
author: S. A. Gusarenko
published in: Russian Mathematics
publication date: 2000-12-07T00:00:00Z
review text: It is known that there is a wide class of the variational problems for which there are no methods of the classical variational calculus to solve them. To solve such problems one has to elaborate the method of reduction of these problems to the minimization problem of some functional in some Hilbert space [see \textit{A. A. Gruzdev}, Izv. Vyssh. Uchebn. Zaved., Mat. 1993, No. 5(372), 46-42 (1993; Zbl 0866.49027) and \textit{A. A. Gruzdev} and \textit{S. A. Gusarenko}, ibid. 1994, No. 6(385), 39-49 (1994; Zbl 0846.49006)]. The author presents a general approach to such reduction, which is based on the direct examination of the extremal problems. The approach is illustrated by the investigation, among others, of the following problems: \[ \textstyle{{1\over 2}} \langle x,Ux\rangle- \langle x,f\rangle\to \inf,\;\ell x= a, \] where \(U: D\to D\) (\(D\) is a Hilbert space) is a linear, self-conjugate and bounded operator, the components of the vector-valued functional \(\ell\) are linearly independent, \(a\in R^k\) and \(f\in D\).
reviewed by: Leon Mikołajczyk
Mathematics Subject Classification ID: 49K27
zbMATH DE Number: 1541612
zbMATH Keywords: quadratic functional
zbMATH Keywords: reduction
zbMATH Keywords: minimization problem
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1541612