Theory And Application Of Tensor Z-Eigenvalues And Tensor Variational Inequalitie

With the advent of big data,the tensor being a multidimensional array has become a powerful tool for data analysis.The spectral theory of tensors is an important part of tensor analysis,which plays an important role in numerical multilinear algebra and automatic control.In addition,the tensor variational inequality is a crucial problem in optimization theory,and the tensor variational inequality is a natural extension of the vector variational inequality,which is widely used in socio-economic analysis and game theory.Therefore,this paper investigates the theory and applications of the tensor Z-eigenvalue problem and the tensor variational inequality problem.The main contents are divided into the following two areas:1.Several new properties of symmetric decomposable tensor Z-eigenvalues in multilinear dynamical systems are given.Under standard orthogonal conditions,the conjecture that the spectral radius of an orthogonal decomposable tensor is included in its coefficients is verified.In addition,several new properties about eigenvectors of symmetric decomposable tensors are proved.As an application of conjecture,a sufficient region is established to ensure the asymptotic stability of the multilinear dynamical system,and its performance is verified by numerical examples.2.Recall the current conclusions about solutions of tensor variational inequality problems,and then provide a new existence result under mild conditions.Combined the inertial contraction projection algorithm with the Mann-type method,a new algorithm is proposed to solve the tensor variational inequality under certain monotonicity or continuity conditions.Furthermore,the paper considers the equilibrium problem of dynamic(i.e.time-dependent)oligopolistic market with over product and demand.Then,an equivalent relationship between tensor variational inequality and the proposed equilibrium problem is proved.Finally,through numerical experiments to verify the effectiveness of the proposed algorithm in solving the above equilibrium problem.