张量分析和多项式优化的若干进展

Some advances in tensor analysis and polynomial optimization

张量分析(也称多重数值线性代数)主要包括张量分解和张量特征值的理论和算法,多项式优化主要包括目标和约束均为多项式的一类优化问题的理论和算法.主要介绍这两个研究领域中若干新的研究结果.对张量分析部分,主要介绍非负张量H-特征值谱半径的一些性质及求解方法,还介绍非负张量最大(小)Z-特征值的优化表示及其解法;对多项式优化部分,主要介绍带单位球约束或离散二分单位取值、目标函数为齐次多项式的优化问题及其推广形式的多项式优化问题和半定松弛解法.最后对所介绍领域的发展趋势做了预测和展望.

Tensor analysis （also called as numerical multilinear algebra） mainly in- cludes tensor decomposition, tensor eigenvMue theory and relevant algorithms. Polynomi- al 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 prop- erties 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 in- troduce the optimization of homogeneous polynomial function under the unit spherical constraints or binary constraints and their extended problems, and semidefinite relax- ation methods for solving them approximately. We also look into the further perspective of these research topics.