Abstract:  Tensor analysis (also called as numerical multilinear algebra) mainly includes tensor decomposition, tensor eigenvalue theory and relevant algorithms. Polynomial optimization mainly includes theory and algorithms for solving optimization problems with polynomial objects functions under polynomial constrains. This survey covers the most of recent advances  in these two fields. For tensor analysis, we introduce some properties and  algorithms concerning the spectral radius of nonnegative tensors' H-eigenvalue. We also discuss the optimization models and solution methods of nonnegative tensors' largest (smallest) Z-eigenvalue. For polynomial optimization problems, we mainly introduce the optimization of homogeneous polynomial function under the unit spherical constraints or binary constraints and their extended problems, and  semidefinite relaxation methods for solving them approximately. We also look into the further perspective of these research topics.

Key words: tensor, eigenvalue, spectral radius, polynomial optimization, algorithm, semidefinite relaxation, approximation algorithm