Generalized Perron Roots and Solvability of the Absolute Value Equation
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arXiv
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| Format: | Preprint |
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2019
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| author | Radons, Manuel Tonelli-Cueto, Josué |
| author_facet | Radons, Manuel Tonelli-Cueto, Josué |
| contents | Let $A$ be a $n\times n$ real matrix. The piecewise linear equation system $z-A\vert z\vert =b$ is called an absolute value equation (AVE). It is well-known to be equivalent to the linear complementarity problem. Unique solvability of the AVE is known to be characterized in terms of a generalized Perron root called the sign-real spectral radius of $A$. For mere, possibly non-unique, solvability no such characterization exists. We narrow this gap in the theory. That is, we define the concept of the aligned spectrum of $A$ and prove, under some mild genericity assumptions on $A$, that the mapping degree of the piecewise linear function $F_A:\mathbb{R}^n\to\mathbb{R}^n\,, z\mapsto z-A\lvert z\rvert$ is congruent to $(k+1)\mod 2$, where $k$ is the number of aligned values of $A$ which are larger than $1$. We also derive an exact--but more technical--formula for the degree of $F_A$ in terms of the aligned spectrum. Finally, we derive the analogous quantities and results for the LCP. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1912_08157 |
| institution | arXiv |
| publishDate | 2019 |
| record_format | arxiv |
| spellingShingle | Generalized Perron Roots and Solvability of the Absolute Value Equation Radons, Manuel Tonelli-Cueto, Josué Optimization and Control Numerical Analysis 65K99, 90C33, 15A24 Let $A$ be a $n\times n$ real matrix. The piecewise linear equation system $z-A\vert z\vert =b$ is called an absolute value equation (AVE). It is well-known to be equivalent to the linear complementarity problem. Unique solvability of the AVE is known to be characterized in terms of a generalized Perron root called the sign-real spectral radius of $A$. For mere, possibly non-unique, solvability no such characterization exists. We narrow this gap in the theory. That is, we define the concept of the aligned spectrum of $A$ and prove, under some mild genericity assumptions on $A$, that the mapping degree of the piecewise linear function $F_A:\mathbb{R}^n\to\mathbb{R}^n\,, z\mapsto z-A\lvert z\rvert$ is congruent to $(k+1)\mod 2$, where $k$ is the number of aligned values of $A$ which are larger than $1$. We also derive an exact--but more technical--formula for the degree of $F_A$ in terms of the aligned spectrum. Finally, we derive the analogous quantities and results for the LCP. |
| title | Generalized Perron Roots and Solvability of the Absolute Value Equation |
| topic | Optimization and Control Numerical Analysis 65K99, 90C33, 15A24 |
| url | https://arxiv.org/abs/1912.08157 |