Abstract
A residual existence theorem for linear equations is proved: if \({A \in \mathbb{R}^{m\times n}}\), \({b \in \mathbb{R}^{m}}\) and if X is a finite subset of \({\mathbb{R}^{n}}\) satisfying \({{\rm max}_{x \in X}p^T(Ax-b) \geq 0}\) for each \({p \in \mathbb{R}^{m}}\), then the system of linear equations Ax = b has a solution in the convex hull of X. An application of this result to unique solvability of the absolute value equation Ax + B|x| = b is given.
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Supported by the Czech Republic Grant Agency under grants 201/09/1957 and 201/08/J020, and by the Institutional Research Plan AV0Z10300504.
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Rohn, J. A residual existence theorem for linear equations. Optim Lett 4, 287–292 (2010). https://doi.org/10.1007/s11590-009-0160-7
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DOI: https://doi.org/10.1007/s11590-009-0160-7