题目：对偶马尔科夫链与带非负标准项的对偶数矩阵

演讲人：祁力群教授，香港理工大学荣休教授

主持人：林贵华教授，上海大学管理学院

时间：2023年10月17日（周二），下午15:30

地点：上海大学校本部东区管理学院467室

主办单位：上海大学管理学院、上海大学管理学院青年教师联谊会

演讲人简介：

国际知名优化专家，香港理工大学荣休教授，俄罗斯Petrovskaya科学与艺术研究院外籍院士，中国运筹学会首届会士。

中国运筹学会科学技术奖一等奖获得者，十种国际期刊的主编或编委。

连续多年入选世界高被引科学家，入选 2021 年全球前2%顶尖科学家榜单。

演讲内容简介：

We propose a dual Markov chain model to accommodate probabilities as well as perturbation, or error bounds, or variances, in the Markov chain process. This motivates us to extend the Perron-Frobenius theory to dual number matrices with primitive and irreducible nonnegative standard parts. We show that such a dual number matrix always has a positive dual number eigenvalue with a positive dual number eigenvector. The standard part of this positive dual number eigenvalue is larger than or equal to the modulus of the standard part of any other eigenvalue of this dual number matrix. We present an explicit formula to compute the dual part of this positive dual number eigenvalue. The Collatz minimax theorem also holds here. The results are nontrivial as even a positive dual number matrix may have no eigenvalue at all. An algorithm based upon the Collatz minimax theorem is constructed. The convergence of the algorithm is studied. We give an upper bound on the distance of stationary states between the dual Markov chain and the perturbed Markov chain. Numerical results on both synthetic examples and dual Markov chain including some real world examples are reported.

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