Research On Tensor Methods Of Polynomial Optimization Problems And Generalized Polynomial Complementarity Problems

Recently,with the advent of the era of big data,tensors and tensor-based methods have gradually appeared in people’s field of vision.Polynomials are closely related to tensors.Each polynomial can be represented by a tensor.The polynomial optimization problem is one of important optimization problems,which has been studied a lot in recent years with rich theory and wide applications.Under certain conditions,the polynomial optimization problem can be transformed into a special complementary problem: generalized polynomial complementary problem,through KKT conditions.This paper mainly studies the polynomial optimization problem and the generalized polynomial complementarity problem via tensor-based methods.Specific contents are as follows:First,we extend the matrix semi-definite programming to the third-order tensor space.With the help of a new multiplication rule between third-order tensors: tensor Tproduct,we systematically discuss the positive semi-definiteness of third-order symmetric tensors,and extend some properties of positive semi-definite matrices to the third-order case.Based on the property that the set composed of all third-order T-positive semidefinite tensors is non-empty,pointed,closed,convex and self-dual cone,we propose the model of semi-definite programming over third-order tensor space.Furthermore,its duality theory is discussed.After that,we provide a simple way to deal with the semi-definite programming over third-order tensor space by transforming it into matrix semi-definite programming over complex field.Last,we show some applications of this new thirdorder tensor semi-definite programming.Second,we study third-order tensor semi-definite programming relaxation for unconstrained polynomial optimization problems.For solving polynomial optimization problems,an important method is matrix semi-definite programming relaxation.The major difficulty brought by this relaxation method is that the scale of semi-definite programming will increase significantly as the degree of polynomials increases or the number of variables increases,which greatly limits the solving of large-scale polynomial optimization problems.In this paper,the moment matrix is extended to obtain a third-order T-moment tensor,and in the same spirit of Lasserre’s relaxation,a sequence approximation method based on tensor semi-definite programming is given to solve unconstrained polynomial optimization problems.As far as the unconstrained block-circulant polynomial optimization problem is concerned,this new proposed method based on tensor semi-definite programming can greatly reduce the size of the semi-definite programming problem that needs to be solved in the end.Third,we discuss the nonemptiness and compactness of solution sets to generalized polynomial complementarity problems,and the existence and uniqueness of solutions of generalized polynomial complementarity problems,with the help of studies on the tensorpair corresponding to leading terms in the polynomial-pair.By using the tool of degree theory,we build bridges between the generalized polynomial complementarity problem and the tensor complementarity problem,and between the generalized polynomial complementarity problem and the polynomial complementarity problem.As a result,the study on the nonemptiness and compactness of solution sets to generalized polynomial complementarity problems can be transformed into the investigation of these two classes of simple problems.It is pointed out that “strong monotonicity of function pairs” cannot be satisfied for a lot of polynomial pairs,which is an important condition for guaranteeing the uniqueness of solutions to generalized complementarity problems.Later,under a broader condition: “strict monotonicity of function pairs”,the global uniqueness and solvability for generalized polynomial complementarity problems are established with some additional assumptions.In addition,several results on global error bounds for a subclass of generalized polynomial complementarity problems: tensor complementarity problems are given.