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<p>In this paper, we extend Hladík’s work [SIAM Journal on Matrix Analysis and Applications 44 (2023) 175–195] on the properties of the solution set of absolute value equations and the associated matrix classes. We revisit the key open questions posed by Hladík for absolute value equations, originally formulated for <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(Ax + |x| = b\)</EquationSource> </InlineEquation>, and address them for the form of absolute value equations considered in this paper, <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(Ax - |x| = b\)</EquationSource> </InlineEquation>. Additionally, this study provides further insights into the convexity of the solution set of absolute value equations and the corresponding linear complementarity problem. We present new results on the solvability of absolute value equations when the matrix <i>A</i> is constrained to special matrix classes, as well as when the right-hand side vector <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\( b \)</EquationSource> </InlineEquation> is nonnegative or nonpositive.</p>

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On the properties of solution set of absolute value equations

  • 期刊论文

  • 发表日期：

    2025年10月27日

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  • Shweta Yadav,
  • Dipti Dubey

摘要

In this paper, we extend Hladík’s work [SIAM Journal on Matrix Analysis and Applications 44 (2023) 175–195] on the properties of the solution set of absolute value equations and the associated matrix classes. We revisit the key open questions posed by Hladík for absolute value equations, originally formulated for \(Ax + |x| = b\) , and address them for the form of absolute value equations considered in this paper, \(Ax - |x| = b\) . Additionally, this study provides further insights into the convexity of the solution set of absolute value equations and the corresponding linear complementarity problem. We present new results on the solvability of absolute value equations when the matrix A is constrained to special matrix classes, as well as when the right-hand side vector \( b \) is nonnegative or nonpositive.