Eigenvalue analysis of constrained minimization problem for homogeneous polynomial

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 strictly copositive if and only if every Pareto H-eigenvalue (Z-eigenvalue) of \({\mathcal {A}}\) is positive, and \({\mathcal {A}}\) is copositive if and only if every Pareto H-eigenvalue (Z-eigenvalue) of \({\mathcal {A}}\) is non-negative.