Extremal problems on the set of nonnegative definite matrices
type: Item
instance of: scholarly article
title: Extremal problems on the set of nonnegative definite matrices
zbMATH Open document ID: 0565.90056
DOI: 10.1016/0024-3795(85)90182-X
published in: Linear Algebra and its Applications
publication date: 1985-01-01T00:00:00Z
review text: The problem of minimizing a function of symmetric matrix elements, subject to the matrix being positive semi-definite, is formulated as a semi-infinite program. Optimality conditions are then stated under the usual assumptions (convexity, Slater's condition). For the linear program with the objective function being the inner product \((A,X)=tr AX\) and the constraints \(S+X\) being positive semi-definite matrices (S is fixed symmetric), conditions are given for uniqueness of the solution and differentiability of the optimal value function. The corresponding duality results recover some of \textit{I. Olkin} and \textit{F. Pukelsheim}'s results [ibid. 48, 257-263 (1982; Zbl 0527.60015)] as special cases.
Mathematics Subject Classification ID: 90C25
Mathematics Subject Classification ID: 90C34
Mathematics Subject Classification ID: 15A39
Mathematics Subject Classification ID: 15A45
Mathematics Subject Classification ID: 90C55
Mathematics Subject Classification ID: 90C31
zbMATH DE Number: 3902405
zbMATH Keywords: trace of a matrix
zbMATH Keywords: Optimality conditions
zbMATH Keywords: positive semi-definite matrices
zbMATH Keywords: uniqueness of the solution
zbMATH Keywords: differentiability of the optimal value function
zbMATH Keywords: duality results
Wikidata QID: Q93582777
author: Alexander Shapiro
reviewed by: Sanjo Zlobec
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3902405