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arXiv · 1302.6085

Eigenvalue analysis of constrained minimization problem for homogeneous polynomial

Yisheng Song·Liqun Qi
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Abstract

In this paper, the concepts of Pareto $H$-eigenvalue and Pareto $Z$-eigenvalue are introduced for studying constrained minimization problem and the necessary and sufficient conditions of such eigenvalues are given. It is proved that a symmetric tensor has at least one Pareto $H$-eigenvalue (Pareto $Z$-eigenvalue). Furthermore, the minimum Pareto $H$-eigenvalue (or Pareto $Z$-eigenvalue) of a symmetric tensor is exactly equal to the minimum value of constrained minimization problem of homogeneous polynomial deduced by such a tensor, which gives an alternative methods for solving the minimum value of constrained minimization problem. In particular, a symmetric tensor $\mathcal{A}$ is copositive if and only if every Pareto $H$-eigenvalue ($Z-$eigenvalue) of $\mathcal{A}$ is non-negative.

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Yisheng Song, Liqun Qi. 2013-02-25. Eigenvalue analysis of constrained minimization problem for homogeneous polynomial. https://doi.org/10.1007/s10898-015-0343-y

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