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, theminimumPareto 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 A is strictly copositive if and only if every Pareto H -eigenvalue (Z -eigenvalue) ofA is positive, andA is copositive if and only if every Pareto H -eigenvalue (Z -eigenvalue) of A is non-negative.