Eigenvalue analysis of constrained minimization problem for homogeneous polynomial

Yisheng Song () and Liqun Qi ()

Journal of Global Optimization, 2016, vol. 64, issue 3, 563-575

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 $${\mathcal {A}}$$ A is strictly copositive if and only if every Pareto H-eigenvalue (Z-eigenvalue) of $${\mathcal {A}}$$ A is positive, and $${\mathcal {A}}$$ A is copositive if and only if every Pareto H-eigenvalue (Z-eigenvalue) of $${\mathcal {A}}$$ A is non-negative. Copyright Springer Science+Business Media New York 2016

Keywords: Constrained minimization; Principal sub-tensor; Pareto H-eigenvalue; Pareto Z-eigenvalue; 15A18; 15A69; 90C20; 90C30; 35P30 (search for similar items in EconPapers)
Date: 2016
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Citations: View citations in EconPapers (8)

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DOI: 10.1007/s10898-015-0343-y

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