On an interconnection between the Lipschitz continuity of the solution map and the positive principal minor property in linear complementarity problems over Euclidean Jordan algebras
type: Item
instance of: scholarly article
title: On an interconnection between the Lipschitz continuity of the solution map and the positive principal minor property in linear complementarity problems over Euclidean Jordan algebras
zbMATH Open document ID: 1138.90033
published in: Linear Algebra and its Applications
publication date: 2007-09-13T00:00:00Z
review text: Let \(V\) be an Euclidean Jordan algebra, \(K\) be a symmetric cone in \(V\), \(L: V\longrightarrow V\) be a linear transformation and \(q\in V\). The linear complementary problem associated to \(L\) and \(q\), \(\text{LCP}(L,q)\), prescribes finding \(x\in V\) such that \(x\in K\), \(Lx+q\in K\) and \(\langle x,Lx+q\rangle=0\). It is well known that when \(V=\mathbb R^n\) and \(L\) is a real matrix, \(\text{LCP}(L,q)\) has a unique solution for all \(q\in \mathbb R^n\), iff all the principal minors of \(L\) are positive. In this case the solution map of the \(\text{LCP}(L,q)\) is well defined and Lipschitz continuous in \(\mathbb R^n\). The main result of this paper establishes one direction of the analogous property in the general case: if the solution map is Lipschitz continuous and if \(L\) has the \(Q\)-property, then \(L\) has the positive principal minor property.
reviewed by: Lyonell Boulton
Mathematics Subject Classification ID: 90C33
Mathematics Subject Classification ID: 17C30
Mathematics Subject Classification ID: 17C50
zbMATH DE Number: 5190762
zbMATH Keywords: complementarity
zbMATH Keywords: positive principal minor property
zbMATH Keywords: Lipschitzian property
zbMATH Keywords: Q-property
author: R. Balaji
MaRDI profile type: Publication
full work available at URL: j.laa.2007.04.004
OpenAlex ID: W1984315737
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DOI: 10.1016/J.LAA.2007.04.004
description: scientific article; zbMATH DE number 5190762