Continuity of the optimal value function in indefinite quadratic programming
type: Item
instance of: scholarly article
title: Continuity of the optimal value function in indefinite quadratic programming
zbMATH Open document ID: 1015.90061
DOI: 10.1023/A:1014011906646
author: N. N. Tam
published in: Journal of Global Optimization
publication date: 2002-05-28T00:00:00Z
review text: Topic of the paper is the continuity and semicontinuity of the optimal value function \[ (D,A,c,b)\to\varphi(D,A,c,b) \] of the quadratic programming problem \[ \text{minimize } c^{T}x+\tfrac{1}{2}x^{T}Dx \;\text{s.t.} Ax\geq b, \] where, in contrast to previous results of other authors, \(D\) is an arbitrary symmetric and hence not necessarily positive semidefinite matrix. The continuity of \(\varphi\) at \((D,A,c,b)\) is completely characterized by the Slater condition for the system \(Ax\geq b\) (C1) and the condition that \(x=0\) is the only solution of the problem for \(c=0\) and \(b=0\) (C2). Moreover, the conditions (C1) and (C2) are shown to be sufficient but not necessary for the upper and lower semicontinuity of \(\varphi\) at \((D,A,c,b)\) respectively. It is finally verified that the sufficient assumptions of \textit{M. J. Best} and \textit{B. Ding} [J. Optim. Theory Appl. 86, 245-250 (1995; Zbl 0838.90118)] for the lower semicontinuity and the continuity of \(\varphi\) are somewhat weaker than the corresponding assumptions of the author, but they only apply to the case of positive semidefinite \(D\) and, as is shown by examples, do not suffice to guarantee the respective property for general indefinite matrices \(D\).
reviewed by: Rembert Reemtsen
Mathematics Subject Classification ID: 90C20
zbMATH DE Number: 1746310
zbMATH Keywords: quadratic programming
zbMATH Keywords: optimal value function
zbMATH Keywords: lower semicontinuity
zbMATH Keywords: upper semicontinuity
zbMATH Keywords: continuity
MaRDI profile type: Publication
full work available at URL: a:1014011906646
OpenAlex ID: W1501389859
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description: scientific article; zbMATH DE number 1746310