Eigenvalue analysis of constrained minimization problem for homogeneous polynomial (Q280985)

The paper is dedicated to the study of constrained minimization problems and, using the concepts of the Pareto \(H\)-eigenvalue and the Pareto \(Z\)-eigenvalue, to the derivation of the necessary and sufficient conditions of such eigenvalues. It is shown that ``a symmetric tensor has at least one Pareto \(H\)-eigenvalue (Pareto \(Z\)-eigenvalue)'' and that ``the minimum Pareto \(H\)-eigenvalue (or Pareto \(Z\)-eigenvalue) of a symmetric tensor is exactly equal to the minimum value of the constrained minimization problem of the homogeneous polynomial deduced by such a tensor''. This result may be used as ``an alternative method for solving the minimum value of the constrained minimization problem''. ``As a corollary, a symmetric tensor \(\mathcal A\) is copositive if and only if every Pareto \(H\)-eigenvalue (\(Z\)-eigenvalue) of \(\mathcal A\) is nonnegative and \(\mathcal A\) is strictly copositive if and only if every Pareto \(H\)-eigenvalue (\(Z\)-eigenvalue) of \(\mathcal A\) is positive.''