报告题目: Least element of nonnegative solution set of a class of polynomial inequalities

报告人:黄正海教授（天津大学）

报告摘要：We consider the least element problem of the nonnegative solution set of a system of inequalities defined by the homogeneous polynomials. In the set under consideration, the homogeneous polynomials are defined by a tensor. When the tensor involved is square, the set under consideration is just the feasible set of the tensor complementarity problem (TCP). We first introduce the concept of the generalized $Z$-type tensor, and prove that it is a generalization of $Z$-tensors even if it reduces to a square tensor. Then, under the conditions that the considered set is nonempty and the tensor involved is a generalized $Z$-type tensor, we propose an iterative method for finding the least element of the considered set. Specifically, we solve a series of corresponding systems of lower-dimensional tensor equations by continuous recognition of the positive components in the least element, and prove that the least element of the set can be obtained within finite step iterations. When the tensor involved is square, the least element obtained is also a solution of the TCP. Compared with the existing methods for finding the least-element solution of the TCP, our method does not require any additional conditions and has lower computational costs. Preliminary numerical experiments show that the proposed method is effective.

报告人简介：黄正海，天津大学数学学院教授、博士生导师。主要从事最优化理论、算法及其应用方面的研究工作，在求解互补与变分不等式问题、对称锥优化与对称锥互补问题、稀疏优化、张量优化、核磁共振医学成像、人脸识别等方面取得了一些有意义的成果。目前的主要研究兴趣是张量优化、特殊结构的变分不等式与互补问题、以及机器学习中的优化理论方法及其应用。已发表SCI检索论文120多篇、连续获得多项国家自然科学基金资助。曾获得中科院优秀博士后奖和教育部高等学校自然科学奖二等奖。目前为中国运筹学会常务理事；国际期刊《Pacific Journal of Optimization》、《Applied Mathematics and Computation》和《Optimization,Statistics & Information Computing》的编委、中国核心期刊《运筹学学报》的编委。

报告时间：2022年1月6日 周四19:00-20:00

腾讯会议：790832712

学院联系人：陈中明

欢迎感兴趣的老师和同学参加！