A note on generalised linear complementarity problems

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Bibliographische Detailangaben
Zeitschriftentitel:	Bulletin of the Australian Mathematical Society
Personen und Körperschaften:	Parida, J., Sahoo, B.
In:	Bulletin of the Australian Mathematical Society, 18, 1978, 2, S. 161-168
Format:	E-Article
Sprache:	Englisch
veröffentlicht:	Cambridge University Press (CUP)
Schlagwörter:	General Mathematics

Details
Zusammenfassung:	<jats:p>Given an<jats:italic>n</jats:italic>×<jats:italic>n</jats:italic>matrix<jats:italic>A</jats:italic>, an<jats:italic>n</jats:italic>-dimensional vector<jats:italic>q</jats:italic>, and a closed, convex cone<jats:italic>S</jats:italic>of<jats:italic>R<jats:sup>n</jats:sup></jats:italic>, the generalized linear complementarity problem considered here is the following: find a<jats:italic>z</jats:italic>∈<jats:italic>R<jats:sup>n</jats:sup></jats:italic>such that</jats:p><jats:p><jats:disp-formula><jats:graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" mimetype="image" position="float" xlink:type="simple" xlink:href="S000497270000798X_eqnU1" /></jats:disp-formula></jats:p><jats:p>where<jats:italic>s</jats:italic>* is the polar cone of<jats:italic>S</jats:italic>. The existence of a solution to this problem for arbitrary vector<jats:italic>q</jats:italic>has been established both analytically and constructively for several classes of matrices<jats:italic>A</jats:italic>. In this note, a new class of matrices, denoted by<jats:italic>J</jats:italic>, is introduced.<jats:italic>A</jats:italic>is a<jats:italic>J</jats:italic>-matrix if</jats:p><jats:p><jats:disp-formula><jats:graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" mimetype="image" position="float" xlink:type="simple" xlink:href="S000497270000798X_eqnU2" /></jats:disp-formula></jats:p><jats:p>The new class can be seen to be broader than previously studied classes. We analytically show that for any<jats:italic>A</jats:italic>in this class, a solution to the above problem exists for arbitrary vector<jats:italic>q</jats:italic>. This is achieved by using a result on variational inequalities.</jats:p>
Umfang:	161-168
ISSN:	0004-9727 1755-1633
DOI:	10.1017/s000497270000798x