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In this paper, the concepts of Pareto $H$-eigenvalue and Pareto$Z$-eigenvalue are introduced for studying constrained minimization problem andthe necessary and sufficient conditions of such eigenvalues are given. It isproved 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 ofconstrained minimization problem of homogeneous polynomial deduced by such atensor, which gives an alternative methods for solving the minimum value ofconstrained 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.

Eigenvalue analysis of constrained minimization problem for homogeneous polynomial