Tensor Co-positivity Determination And Related Analysi

Copositive tensor is a kind of important structured tensor,which has broad applications in unform hypergraph theory,Higgs vacuum stability,tensor spectral theory,standard Bi-quadratic programming and so on.During the application process,detecting tensor copositivity and the large scale copositive tensor optimization problem are two challenging problems.Based on this,this thesis studies the copositive tensor and related optimization problems.This thesis aims to study the approximation of higher order copositive tensor cone,copositivity detection and optimization methods for solving large scale copositive tensor optimization problems.In Chapter 1,we briefly introduce the research status,scientific significance and related basic knowledge of copositive tensor and copositive tensor optimization at home and abroad.In Chapter 2,we consider the copositive tensor optimization problem(CTOP)with high-order copositive tensors.Firstly,we prove that the given tensor is strictly copositive if and only if the Slater constraint holds.Then,based on the standard simplex,two new inner and outer approximations of the copositive tensor cone are established.Through these approximations,a linear approximation algorithm for solving copositive tensor programming is proposed and its convergence is proved.Finally,the proposed algorithm is applied to estimate the coclique number of uniform hypergraphs,and numerical results show its efficiency.In Chapter 3,an outer approximation of a copositive tensor cone and an inner approximation of a completely positive tensor cone are given.First,based on the property of the copositive tensor,we transform the copositive tensor into a nonnegative polynomial problem.Based on the nonnegativity theory of homogeneous polynomial forms on closed sets,a simple interpretation and asymptotically convergent outer approximation hierarchies of the copositive cone is given,and the corresponding dual cone forms an inner approximating convergence approximation hierarchies the completely positive tensor cone.In Chapter 4,we consider the problem of detecting the copositivity of partially symmetric rectangular tensors.Firstly,based on the equivalence conditions of partially symmetric copositive tensors,the problem of determining the copositive property of partially symmetric tensors is equivalent transformed into a polynomial optimization problem.Secondly,a semidefinite relaxation method for solving polynomial optimization problems is given by Lasserre relaxation method and its convergence is discussed.Several preliminary numerical results confirm our theoretical findings.In Chapter 5,the approximation of the partially symmetric copositive tensor cone and its application in standard Bi-quadratic(StBQP)programming are studied.Firstly,based on the theory of standard simplex,two new inner and outer approximations of the partially symmetric copositive tensor cone are established.Secondly,because the standard Bi-quadratic programming problem is equivalent to a partially symmetric copositive tensor programming problem,that is,the linear cone optimization problem on the cone of partially symmetric copositive tensors,a linear approximation algorithm for solving the partially symmetric copositive tensor programming is given by using the obtained approximation.Finally,numerical results confirm our theoretical findings.In this thesis,we make full use of the special structure of the(partially symmetric)copositive tensor cone and its dual cone generators,and study the approximation of the copositive tensor cone and its dual cone,which provides theoretical support for the research and application of the copositive tensor programming.