Some Studies On Optimization Problems Over Tensor Spaces

In the era of big data,problems in reality are becoming more and more sophisticated,so in recent years it has become a hot topic that how to build better mathematical models to solve the increasingly complex real-world problems.Recently,the tensor(or hypermatrix),as an extension of the matrix,has drawn widespread attention and has become an efficient tool to describe complicated data.Using vectors and matrices as variables no longer satisfies the requirements for modeling some practical problems,so it is necessary to model these problem with tensor variables.Inspired by these,we make some studies on optimization problems over tensor spaces,which is stated as follows:Firstly,we introduce a contraction product of tensors,which is a generalization of the mode product between tensors and vectors.We do research on some basic properties of this tensor product,such as the scalar multiplication,the commutative law,the associative law and the distributive law;discuss the associated positive semidefinite tensors,the gradient and monotonicity of the associated quadratic tensor functions;besides,we study the properties of tensor product with different structure tensors.Secondly,with the aid of the introduced contraction product of tensors,we define a class of affine variational inequalities over tensor spaces.Then,we discuss some properties of the solution set of the affine variational inequality,including existence and uniqueness of the solution and boundedness of the solution set;besides,we investigate a class of oligopolistic market games and transform it into an affine variational inequality over a tensor space.We also define a class of linear complementarity problems over tensor spaces.We discuss an equivalent model of the problem,the feasibility and solvability theory of the problem,and convexity of the solution set;and then,we propose an extragradient method for solving the linear complementarity problem over tensor spaces,and show the convergence of the method under suitable assumptions.The preliminary numerical results are also reported.Finally,we investigate the generalized tensor function over the third-order real tensor space and show that the generalized tensor function can inherit a lot of good properties from the associated scalar function,including continuity,directional differentiability,Frechet differentiability,Lipschitz continuity and semismoothness.These properties provide an important theoretical basis for the studies of various tensor optimization problems with generalized tensor functions.