优化系列报告1
题目：Tensor eigenvalue complementarity problem and tensor eigenvalues
报告人：张立平教授（清华大学）
时间：2020年11月18日20：00-22：00
腾讯会议ID： 110 328 808
摘要： In this talk, we focus on establishing an algorithm to solve the tensor eigenvalue complementarity problem (TEiCP). First, a smoothing Newton-type algorithmis proposed for the TEiCP based on the CHKS smoothing function. Its global convergence is established under some mild conditions. Numerical experiments are reported to show that the proposed algorithm is efficient and could detect more solutions than some existing methods. Second, we apply the proposed algorithm to solve the eigenvalue problem of nonnegative tensors. We analyze the relationship between TEiCP and the H-eigenpair and Z-eigenpair problems of an irreducible nonnegative tensor. We show that the TEiCP with an irreducible nonnegative tensor and unit tensor has a unique solution, which is just the unique positive H-eigenpair of the irreducible nonnegative tensor. We also show that the solution set of the TEiCP with an irreducible nonnegative tensor and identity tensor is nonempty and its solutions are positive. Moreover, we can obtain positive Z-eigenpairs of the irreducible nonnegative tensor from these solutions. Finally, we also apply the proposed algorithm to find the unique positive H-eigenpair and a positive Z-eigenpair of an irreducible nonnegative tensor, the numerical results indicate its efficiency and promising performance.
个人简介：张立平，清华大学长聘副教授，博士生导师，研究方向最优化理论算法及应用，在求解互补与变分不等式问题、半无限规划、张量优化等方面取得了一些有意义的结果。在Mathematics of Computation, SIAM Journal on Matrix Analysis and Applicatios, SIAM Journal on Optimization, Journal of Global Optimization, Applied Numerical Mathematics等期刊已发表高质量论文四十余篇、连续获得多项国家自然科学基金资助。曾获得教育部自然科学奖二等奖和北京市科学技术奖二等奖。
