Eigenvalue analysis of constrained minimization problem for homogeneous polynomial

Yisheng Song, Liqun Qi

Research output: Journal article publicationJournal articleAcademic researchpeer-review

20 Citations (Scopus)

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 (Formula presented.) is strictly copositive if and only if every Pareto H-eigenvalue (Z-eigenvalue) of (Formula presented.) is positive, and (Formula presented.) is copositive if and only if every Pareto H-eigenvalue (Z-eigenvalue) of (Formula presented.) is non-negative.
Original languageEnglish
Pages (from-to)563-575
Number of pages13
JournalJournal of Global Optimization
Volume64
Issue number3
DOIs
Publication statusPublished - 1 Mar 2016

Keywords

  • Constrained minimization
  • Pareto H-eigenvalue
  • Pareto Z-eigenvalue
  • Principal sub-tensor

ASJC Scopus subject areas

  • Computer Science Applications
  • Control and Optimization
  • Management Science and Operations Research
  • Applied Mathematics

Cite this