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Keywords:
max-min algebra; interval matrix; strong robustness; AE(EA) robustness
max-min algebra; interval matrix; strong robustness; AE(EA) robustness
Summary:
In max-min algebra the standard pair of operations plus and times is replaced by the pair of operations maximum and minimum, respectively. A max-min matrix $A$ is called strongly robust if the orbit $x,A\otimes x, A^2\otimes x,\dots$ reaches the greatest eigenvector with any starting vector. We study a special type of the strong robustness called the strong \textit{\textbf{X}}-robustness, the case that a starting vector is limited by a lower bound vector and an upper bound vector. The equivalent condition for the strong \textit{\textbf{X}}-robustness is introduced and efficient algorithms for verifying the strong \textit{\textbf{X}}-robustness is described. The strong \textit{\textbf{X}}-robustness of a max-min matrix is extended to interval vectors \textit{\textbf{X}} and interval matrices \textit{\textbf{A}} using for-all-exists quantification of their interval and matrix entries. A complete characterization of AE/EA strong \textit{\textbf{X}}-robustness of interval circulant matrices is presented.
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