Perron eigenvalue优化中的贪婪策略
The greedy strategy in optimizing the Perron eigenvalue
摘要：
本文讨论了一类非负矩阵族上的谱半径的最小化和最大化问题。对于一些特殊的家庭，这些一般难以解决的问题可以得到有效的解决。我们考虑所谓的产品族，其中每个矩阵由独立于给定集合的行组成。最近引入的贪婪方法工作得非常快。然而，它主要适用于严格正矩阵。对于稀疏矩阵，它往往发散，给出错误的结果。本文提出了一种适用于所有非负乘积族（包括稀疏乘积族）的“选择性贪婪法”，证明了该方法的二次收敛速率，并通过数值算例证明了该方法的有效性.数值算例分别针对有限不确定集和由线性不等式给出的多面体不确定集.在维数达到2000的情况下，在几次迭代中就可以找到乘积族中具有最小/最大谱半径的矩阵。应用到动力系统和图论被认为是
英文摘要：
We address the problems of minimizing and of maximizing the spectral radius overa compact family of non-negative matrices. Those problems being hard in generalcan be efficiently solved for some special families. We consider the so-called prod-uct families, where each matrix is composed of rows chosen independently from givensets. A recently introduced greedy method works very fast. However, it is applicablemostly for strictly positive matrices. For sparse matrices, it often diverges and gives awrong answer. We present the "selective greedy method" thatworks equally well forall non-negative product families, including sparse ones.For this method, we provea quadratic rate of convergence and demonstrate its efficiency in numerical examples.The numerical examples are realised for two cases: finite uncertainty sets and poly-hedral uncertainty sets given by systems of linear inequalities. In dimensions up to 2000, the matrices with minimal/maximal spectral radii in product families are foundwithin a few iterations. Applications to dynamical systemsand to the graph theoryare considered
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