In this article, we characterize strong versions of several matrix classes that appear in the linear complementarity problem (LCP) for uncertain data represented by intervals. The properties of the constraint matrix reflect many characteristics of the LCP, including uniqueness, solvability, number of solutions, convexity of the solution set, etc. We discuss some of the computationally more difficult classes. In particular, we discuss the strong version of the class of almost semimonotone, E(d)-matrix, <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13226_2025_833_Article_IEq1.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="36" /> </InlineMediaObject> <EquationSource Format="TEX">${\bar{E}}(d)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mover accent="true"> <mrow> <mi>E</mi> </mrow> <mrow> <mo stretchy="false">¯</mo> </mrow> </mover> <mo stretchy="false">(</mo> <mi>d</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>-matrix, L(d)-matrix, <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13226_2025_833_Article_IEq2.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="34" /> </InlineMediaObject> <EquationSource Format="TEX">${\bar{L}}(d)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mover accent="true"> <mrow> <mi>L</mi> </mrow> <mrow> <mo stretchy="false">¯</mo> </mrow> </mover> <mo stretchy="false">(</mo> <mi>d</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>-matrix, and <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13226_2025_833_Article_IEq3.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="22" /> </InlineMediaObject> <EquationSource Format="TEX">$Q_{0}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>Q</mi> <mn>0</mn> </msub> </math></EquationSource> </InlineEquation>-matrix. Finally, we introduce positive subdefinite interval matrices.

Additional classes of interval matrices in linear complementarity theory

期刊论文
发表日期：

2025年7月12日

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Gambheer Singh,
Sajal Ghosh,
S. K. Neogy

摘要

In this article, we characterize strong versions of several matrix classes that appear in the linear complementarity problem (LCP) for uncertain data represented by intervals. The properties of the constraint matrix reflect many characteristics of the LCP, including uniqueness, solvability, number of solutions, convexity of the solution set, etc. We discuss some of the computationally more difficult classes. In particular, we discuss the strong version of the class of almost semimonotone, E(d)-matrix, ${\bar{E}}(d)$ $\bar{E} (d)$ -matrix, L(d)-matrix, ${\bar{L}}(d)$ $\bar{L} (d)$ -matrix, and $Q_{0}$ $Q_{0}$ -matrix. Finally, we introduce positive subdefinite interval matrices.